Geometry – from the Greek; geo meaning earth and metry ...
Chapter 1-1 Using Patterns and Inductive Logic
Geometry – from the Greek; geo meaning earth and metry meaning measures.
Ordinary plane geometry generally deals with the application of definitions, postulates and theorems and is based on Euclid’s work, Elements, from about 300 B.C.
Branch of mathematics that involves the measurements, properties and relationships of all shapes and sizes of things (triangles, circles, rectangles, cones, spheres, …)
Inductive reasoning – reach conclusions based on a pattern of specific examples or past events
Patterns, sequences and series – using inductive reasoning to come up with the next entry
Logic – we assume Postulates, make Conjectures, and prove Theorems
Postulate – an accepted statement of fact
Conjecture – a conclusion reached by inductive reasoning
Theorem – something we have proven to be true
Arriving at Conjectures and Proving Theorems:
One very good way of coming up with a reasonable conjecture is to look at past experience. We can see what has always happened in the past and guess that the same thing will continue to happen in the future. For example it has never in recorded history been known to snow in North Carolina in August. We can predict that it will not snow in North Carolina this August. Another good way to come up with reasonable conjectures is to look at patterns and try to guess what is coming next.
If I give you the pattern 1,3,5,7,9,11…
How about 3, 6, 9, 12, 15………..
How about 7, 13, 19, 25………..
How about S, M, T, W, T, F……
How about O, T, T, F, F, S,…
When we use a past experience or pattern to predict what is coming next we are using inductive reasoning.
Inductive reasoning:
Reaching conclusions based on a pattern of specific examples or past events.
Much of science is based on inductive reasoning. Scientists do lots of experiments and try to see patterns and try to predict what will happen next or they try, based on their experiments, to get enough experience to say what it is that will usually happen.
Inductive reasoning is very powerful and can lead us to many correct conclusions. However, it is not perfect and sometimes it can lead to incorrect results.
|Connect the points with as many segments as possible. |How many non-overlapping regions are formed? |
|2 points |[pic] | |
|3 points |[pic] | |
|4 points |[pic] | |
|5 points |[pic] | |
|Look at the pattern of non-overlapping regions. What conjecture can you make about how many non-overlapping regions you can make by connecting six |
|points with as many segments as possible? |
|Conjecture ( |
|Use the diagram below to test your conjecture. |
|6 points |[pic] | |
Even conjectures that seem obvious can be false and that is why we try to prove things. We try to prove things using only postulates and previously proven theorems and definitions and logically connecting them. When we have proven something that way we call it a theorem and say that it is always true.
For example I can say 2 + 1 = 3 because the definition of the integers tells me that if I add one I get to the next integer. Most things are not so easy to prove and we will be seeing a lot of theorems in this class and we will be proving many of them.
Chapter 1-2 Points, Lines, and Planes
Point – no size. Can think of as a location. Represent with a dot and label with a capital letter. • A
Space – the set of all points.
Line – a series of points that extends in two opposite directions with no end (extends infinitely). Name a line by choosing two points on the line or by using a lowercase letter.
Collinear – points that lie on the same line.
Plane – a flat surface that extends in all directions with no end. It has no thickness. Planes are named with a single uppercase letter or by identifying 3 non-collinear points.
Coplanar – points and lines that are in the same plane.
Now that we have these basic definitions we are going to get to our first postulates.
Postulate 1:
Through any 2 points there is one and only one line.
Postulate 2:
If 2 lines intersect, then they intersect in exactly one point.
Postulate 3:
If 2 planes intersect they intersect in a line.
Postulate 4:
Through any 3 noncollinear points there is one and only one plane.
Chapter 1-3 Segments, Rays, Parallel Lines, and Planes
Line segment – a piece of a line. We represent line segments by drawing a line and putting dots at each end and labeling the endpoints with capital letters.
Ray – part of line. It has one endpoint and it consists of all the points on the line going in one direction from that endpoint. When we draw a ray we mark the endpoint and we draw a line going from that point in one direction with an arrow at the end to show that it will go on forever in that direction.
Opposite rays – two collinear rays with the same endpoint that together form a line.
Intersecting Lines – Lines that cross each other and meet at one and only one point. Intersecting lines are always coplanar.
Parallel lines – Coplanar lines that never meet.
Skew lines – Lines that are in different planes. They never meet because they are in different planes so are not intersecting and are not in the same plane so cannot be parallel.
Intersecting planes cross each other and meet at a line.
Parallel planes never meet.
Chapter 1-4 Measuring Angles and Segments
The Ruler Postulate (Postulate 1-5)
The points of a line can be put into a one-to-one correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers.
CONGRUENT
Very important concept. Objects that are congruent have the exact same measures. Objects that are congruent are exactly the same size. The symbol that shows two objects are congruent is (.
Two line segments are congruent if they are exactly the same length. So if line segment AB and line segment CD are congruent, we write AB ( CD.
Segment Addition Postulate (Postulate 1-6)
If points A, B, and C are collinear and B is between A and C, then AB + BC = AC.
Angle:
An angle is formed by two rays with a common endpoint. The rays are called the sides and the common endpoint is called the vertex.
An angle can be named several ways:
By a capital letter (A
By a single number (usually placed inside the angle) (3
By identifying three points, one on each ray and the vertex, with the vertex always being the middle point (CDE
Angles are measured in degrees and the measure of an angle is written as m(A. A protractor is used to measure angles using the following postulate:
The Protractor Postulate (Postulate 1-7)
Let rays OA and OB be opposite rays in a plane (thus they are a line). Ray OA and OB and all the rays with endpoint O that can be drawn on one side of line AB can be paired with the real numbers from 0 to 180 in such a way that:
Ray OA is paired with 0 and ray OB is paired with 180
If ray OC is paired with x and ray OD is paired with y,
then m(COD = |x – y|.
Use a protractor to draw angles of 25, 75 and 125 degrees. Also draw a line and draw angles of given degrees off the line (particularly at specific points on the line).
Angles are classified according to their measures. If m(A = x( then the following definitions are true:
• If 0( < x( < 90( then the angle is Acute.
• If x( = 90( then the angle is Right.
• If 90( < x( < 180( then the angle is Obtuse.
• If x( = 180( then the angle is a Straight Angle.
Angle Addition Postulate
If point B is in the interior of (AOC, then m(AOB + m(BOC = m(AOC.
If (AOC is a straight angle, then m(AOB + m(BOC = 180.
Angles are congruent if they have the same degree measure. The lengths of the rays that make up the angle or the orientation of the angle make no difference.
Chapter 1-5 Good Definitions
We have had a lot of definitions already. It’s important that we understand what makes a good definition.
Some examples of definitions:
A computer is a machine that calculates.
A square is a figure with 4 right angles
A nose is the organ of smell in a living creature.
A fish is an animal that does not live on the land and is not a mammal
A tree is a plant that is bigger than other plants.
A good definition has the following characteristics:
• Uses clearly understood terms. (The terms should be commonly understood or previously defined.)
• Is precise. (Avoid ambiguous words such as big, sort of, some.)
• States what the term is, rather than what it is not.
• Is reversible.
One way to test a definition is to look for a counterexample that shows the definition is wrong.
Midpoint – a point that divides a segment into two congruent segments. (a good definition)
Bisector – A bisector is a line, line segment or ray that cuts something in half.
Perpendicular lines – two lines that intersect to form right angles.
Perpendicular bisector – a line, segment or ray that is perpendicular to a segment at its midpoint.
Angle bisector – a ray that divides an angle into two congruent angles.
Chapter 1-6 Basic Constructions
Construction 1: Congruent segments
Given: segment AB
Step 1: Draw a ray with endpoint C
Step 2: Open the compass to the length of AB
Step 3: With the same compass setting, put the compass point on C. Draw an arc that intersects the ray. Label the intersection point D. CD ( AB
Construction 2: Congruent Angles
Given: (A
Step 1: Draw a ray with endpoint S
Step 2: with the compass point on A, draw an arc that intersects the sides of A. Label the points of intersection B and C.
Step 3: With the same compass setting, put the compass point on S. Draw an arc that intersects the ray at point R.
Step 4: Open the compass to the length of BC. Keeping the compass setting, put the compass point on R. Draw an arc to determine point T.
Step 5: Draw ST. (S ( (A
Construction 3: Perpendicular Bisectors
Given: segment AB
Step 1: Open the compass so that it is open at least as far as half the length of AB and not as far as the whole length of AB. Put the compass point on A and draw an arc. Keep the compass setting the same for step 2.
Step 2: Put the compass point on B and draw an arc. Label the points where the arcs intersect as X and Y.
Step 3: Draw segment XY. Label the intersection of AB and XY as M.
M is the midpoint of AB and XY is the perpendicular bisector of AB.
Construction 4: Angle Bisector
Given: (A
Step 1: Put the compass point on vertex A. Draw an arc that intersects the sides of A. Label the point of intersection B and C.
Step 2: put the compass point on C and draw an arc. Keep the same compass setting and repeat with B. Be sure the arcs intersect. Label the point of intersection X.
Step 3: Draw AX. AX is the angle bisector of angle A
Chapter 1-7 Using Deductive Reasoning
To go from a conjecture to a theorem we need to prove the conjecture. One way to do that is deductive reasoning.
Deductive reasoning:
Deductive reasoning is a process of reasoning logically from given facts to a conclusion. If the given facts are true, deductive reasoning always produces a valid conclusion.
You use deductive reasoning every day.
In order to use deductive reasoning we need postulates and definitions, which we all accept as true.
The properties of the real numbers are basic postulates which we accept because they follow from the definitions of numbers. These properties are:
|Reflexive Property |a = a |
|Symmetric Property |If a = b, then b = a |
|Transitive Property |If a = b and b = c, then a = c |
|Addition Property |If a = b, then a + c = b + c |
|Subtraction Property |If a = b, then a – c = b – c |
|Multiplication Property |If a = b, then a( c ) = b( c ) |
|Division Property |If a = b, then a ( c = b ( c ( c ( 0) |
|Substitution Property |If a = b, then b can replace a in any expression |
|Distributive Property |a(b + c) = ab + ac |
There are also properties of congruence, which follow directly from the definition of congruence.
|Reflexive Property |[pic] ; (A ( (A |
|Symmetric Property |if [pic] then [pic] ; if (A ( (B then (B ( (A |
|Transitive Property |if [pic]and [pic]then [pic] ; |
| |if (A ( (B and (B ( (C then (A ( (C |
Special Types of Angle Pairs:
Vertical Angles:
Vertical angles are angles whose sides are opposite rays.
Adjacent Angles:
Adjacent angles are 2 coplanar angles with a common side, a common vertex and no common interior points.
Complementary Angles:
Complementary angles are two angles the sum of whose measures is 90°.
Supplementary Angles:
Supplementary angles are two angles the sum of whose measures is 180°.
Theorem 1: Vertical angles are congruent.
Theorem 2: If 2 angles are supplements of congruent angles (or of the same angle), then the 2 angles are congruent.
Theorem 3: If 2 angles are complements of congruent angles (or of the same angle), then the two angles are congruent.
Using diagrams…
You CAN conclude:
Angles are vertical.
Angles are adjacent.
Angles are adjacent supplementary.
You CANNOT conclude (unless there are marks giving the information):
Angles or segments are congruent.
An angle is a right angle.
Lines are perpendicular or parallel.
Chapter 1-8 The Coordinate Plane
How do we find the distance from point E to point F? From J to G? From A to J? It is easy to find the distance between any 2 points that are on the same vertical or horizontal line.
To find the distance between point G and point H is not so easy. We need a formula:
The Distance Formula:
D = [pic]
This is a very important formula. We will be using it all the time. You don’t have to memorize it because I will always give it to you on tests. However, you must know how to use it. It comes directly from the Pythagorean Theorem.
There is another formula that we will be using all the time. It is the midpoint formula.
The midpoint formula is easy to remember because it makes sense: a midpoint is the middle of 2 things so it is the average between them. Again, you don’t have to memorize it because I will always give it to you on tests.
The Midpoint Formula:
M = [pic], [pic]
If we know the coordinates of the midpoint of a segment and we know one endpoint, we can use the midpoint formula to find the other endpoint.
EX: A line segment AB has endpoint A (2, -4) and midpoint M (3, -5). Find the coordinates of point B.
We will call the coordinates of B (x, y)
Then 3 = [pic] and -5 = [pic]
6 = 2 + x -10 = -4 + y
x = 4 y = -6
Chapter 2-1 Triangles
Theorem 2-1: The sum of the measures of the angles of a triangle is 180.
In any triangle ABC: m(A + m(B + m(C = 180
Definition of Exterior Angle:
An exterior angle of a polygon is an angle formed by a side of the polygon and an extension of another side of the polygon.
Theorem 2-2: The measure of each exterior angle of a triangle equals the sum of the measures of its 2 remote interior angles.
[pic]
Which angles are exterior angles?
Which angles, that are outside the triangle, are not called exterior angles for this triangle?
Why are these angles not exterior angles?
Which angles are congruent?
Which angles are the remote interior angles for angles a, b, c, d, e and f?
Corollary:
A corollary is a statement that follows directly from a theorem. It is something that becomes obvious once you have the theorem.
Corollary to Exterior Angle Theorem: The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles.
Classifying Triangles
Equilateral
An equilateral triangle is a triangle in which all sides are congruent.
Isosceles
An isosceles triangle is a triangle in which at least 2 sides are congruent.
Scalene
A scalene triangle is a triangle in which no sides are congruent.
Equiangular
An equiangular triangle is a triangle in which all angles are congruent.
Acute
An acute triangle is a triangle in which all the angles are acute.
Right
A right triangle is a triangle with one right angle.
Obtuse
An obtuse triangle is a triangle with one obtuse angle
Chapter 2-2 Polygons
Polygon
A polygon is closed plane figure with at least 3 sides. The sides intersect only at their endpoints and no adjacent sides are collinear.
A triangle is a polygon.
To identify a polygon use a capital letter for each vertex, then start at any vertex and list the other vertices consecutively.
Diagonal
A diagonal of a polygon is a line segment that joins any 2 vertices.
Convex
A polygon is convex if no diagonal contains points outside the polygon.
Concave
A polygon is concave if even one diagonal contains points outside the polygon
(A concave polygon “caves in” on itself someplace and a convex one never does.)
You may assume a polygon is convex unless told otherwise.
Polygons are classified by their number of sides.
To find the sum of the angles of a polygon, divide the polygon into triangles – we know each triangle has 180 degrees.
|Classification of Polygons |
|Polygon |Number of Sides |Sum of Interior Angles |
|Triangle |3 |180 |
|Quadrilateral |4 | |
|Pentagon |5 | |
|Hexagon |6 | |
|Heptagon |7 | |
|Octagon |8 | |
|Nonagon |9 | |
|Decagon |10 | |
|Dodecagon |12 | |
|n-gon |n | |
Theorem 2-3: The sum of the of the measures of the interior angles of an n-gon is [pic]
Theorem 2-4: The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.
Why do we have to say one at each vertex?
Why can we pick either one?
Equilateral Polygon
An equilateral polygon is a convex polygon in which all the sides are congruent.
Equiangular Polygon
An equiangular polygon is a convex polygon in which all the angles are congruent.
REGULAR POLYGON
A regular polygon is both equilateral and equiangular
For example, this is a regular hexagon.
[pic]
What does that mean?
The sum of the measures of the interior angles is?
What is the measure of each interior angle?
What is the measure of each exterior angle?
How do we know all the exterior angles are the same?
Chapter 2-3 Parallel and Perpendicular Lines in the Coordinate Plane
Parallel and Perpendicular Lines
What does slope mean in English? – how steep an incline is. It means exactly the same thing in math. If I want to wash the second story windows in my house and they are 12 feet off the ground and I have a very long adjustable ladder, I can put the ladder really close to the house and have a very steep slope or I can put it very far from the house and have a small slope. For the same distance I go up, if I go a small distance horizontally the slope is large and if I make the distance I go “over” (horizontally) large the slope is small. If I am going up a path on a hill and I go up fast compared to how far over I have gone the slope is steep and if I go up slowly compared to how far over I have gone the slope is small.
[pic]
What does a zero slope mean?
What does an undefined slope mean?
Slope
The slope of a line means the rate of vertical change of the line, relative to the rate of horizontal change. It means the rate at which a line goes up (vertical or ‘the rise’), divided by the rate at which the line goes over (horizontal or ‘the run’).
The slope, in math is usually denoted by the letter m.
So [pic] or another way to say this is [pic]
The slope of a line is constant, it can’t change anywhere on the line or it wouldn’t be a line. Why?
Since the slope is constant, if you know the slope of a line and any point on the line, it is very easy to graph the line. Simply start at the point you know and then go up as far as the change in y and over as far as the change in x and you will get to another point on the line. Once you have 2 points you have determined the line.
That’s why it is very easy to plot a line if you are given the equation in slope intercept form. The slope intercept form of the equation of a line is
y = mx + b
Where m stands for the slope and b is the y-intercept. What does the y-intercept mean?
EX: Graph [pic] (use the slope-intercept method)
[pic]
Graph [pic](use the slope-intercept method)
[pic]
Parallel lines
Parallel lines are coplanar lines that never intersect. Parallel lines have the same slope.
EX 1: Write the equations of 3 different lines parallel to y = - 4x + 2
EX 2: Write the equations of 3 different lines parallel to y = 2
EX 3: Write the equations of 3 different lines parallel to x = -2
Graph 2 of your equations from EX 1:
[pic]
All horizontal lines are parallel to each other and all vertical lines are parallel to each other.
That means all lines of the form y = c (where c is a constant) are parallel.
All lines of the form x = c are parallel.
Perpendicular lines
Perpendicular lines are lines that meet at right angles. Perpendicular lines have slopes that are the negative reciprocals of each other.
Why did I not have to specify that perpendicular lines are coplanar?
What does it mean for 2 numbers to be negative reciprocals?
EX 4: Write the equations of 3 different lines perpendicular to y = -4x +2
EX 5: Write the equations of 3 lines perpendicular to y = 2
EX 6: Write the equations of 3 lines perpendicular to x = -2
Graph the equation in EX 4 plus 2 of your equations, here.
All horizontal and all vertical lines are perpendicular to each other.
That means all lines of the form y = c (where c is any constant) are perpendicular to all lines of the form x = a (where a is any constant). Can a equal c?
Theorem 2-5: Two lines parallel to a third line are parallel to each other.
Theorem 2-6: In a plane, two lines perpendicular to a third line are parallel to each other
Chapter 2-4 Classifying Quadrilaterals
What does quadrilateral mean?
Triangle means 3 angles.
Quad means 4.
Lateral means sides.
Quadrilateral means 4 sided.
Classifying Special Quadrilaterals:
Parallelogram
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
Rhombus
A rhombus is a parallelogram with 4 congruent sides.
Rectangle
A rectangle is a parallelogram with 4 right angles.
Square
A square is a parallelogram with 4 congruent sides and 4 right angles.
Kite
A kite is a quadrilateral with 2 pairs of adjacent sides congruent and no opposite sides congruent.
Trapezoid
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Isosceles Trapezoid
An isosceles trapezoid is a trapezoid whose non-parallel sides are congruent.
Special Quadrilaterals and How They Relate to Each Other
[pic]
In order to figure out what kind of quadrilateral you have, you have to be able to determine if opposite sides are parallel, if adjacent sides are perpendicular and if sides are the same length.
How do you find out those things?
To find the length of sides use the distance formula.
To find out if sides are parallel or perpendicular, find the slope.
Review: the slope of a line is [pic]
What makes lines parallel - equal slopes.
What makes lines perpendicular - their slopes are the negative reciprocals of each other.
Which of these lines are perpendicular and which are parallel?
a) y = 2x + 1 b) y = – x c) y = x – 4 d) y = ½ e) y = -2x + 3
f) y = 2x – 5
If you are given a quadrilateral, you probably won’t be given the equations of the lines that make up the sides. You will probably only be given the coordinates of the vertices.
Let’s say 2 of the vertices of a quadrilateral are A(9,-2) and B(3,4). To find the slope of the line connecting them [pic]
If all you want is to see is if it is parallel or perpendicular to some other line you can just compare the slopes.
Reminder: a zero slope means the line is flat – there is no change in y. The equation of a line with zero slope is y = a constant; it can be y = 3 or y = - 7 or y = any number. All of those lines are horizontal. All lines of that form are parallel. They are only perpendicular to vertical lines. Vertical lines have undefined slopes and they are of the form x = a constant; it can be x = 6 or x = -9 or x = any number. Vertical lines are parallel to all other vertical lines.
Try one: (pg 94, 18)
Given: the coordinates of the vertices of a quadrilateral are J(2,1), K(5,4), L(7,2),M(2,-3)
Determine the best name for the quadrilateral
[pic]
Chapter 2-5 Circles
Circle
A circle is the set of all points equidistant from a given point called the center. A capital letter denotes the center of the circle and the circle is named by its center. This circle is named circle O.
[pic]
Radius
A radius is a segment with one end at the center of the circle and the other on the circumference. Name several radii on circle O. By the definition of a circle all radii are congruent.
Diameter
A diameter of a circle is a segment that contains the center of the circle and has both endpoints on the circumference. The length of a diameter is twice the length of a radius. Name a diameter on the diagram.
Central Angle
A central angle is an angle whose vertex is at the center of the circle.
Name a central angle on your worksheet
Arc
An arc is a part of the circumference of the circle.
Name some arcs on the circle.
Semicircle
A semicircle is an arc that is half of a circle.
Every diameter cuts a circle into 2 semicircles.
The measure of a semicircle is 180(.
Minor Arc
A minor arc is an arc that is smaller than a semicircle.
A minor arc is designated by its 2 endpoints on the circumference.
The measure of minor arc is the measure of the central angle whose endpoints on the circumference are the same as the endpoints of the arc.
On the circle we are using AB and CD are both minor arcs. The measure of AB = m(AOB. What angle determines the measure of CD?
Major Arc
A major arc is an arc that is larger than a semicircle.
A major arc is always designated by 3 points on the circumference of the circle - the 2 endpoints and any one point in between them. It is necessary to use 3 points for a major arc to differentiate it from a minor arc. If I only mentioned 2 points on the circumference of the circle, without some convention, you wouldn’t know if I meant go the long way between them or the short way.
On your circle, ADE is a major arc (if I just said AE it would mean the minor arc).
The measure of a major arc is 360( minus the measure of the minor arc that the 2 endpoints make.
Adjacent Arcs
Adjacent arcs are 2 arcs in the same circle that have exactly one point in common.
Which arcs on your worksheet are adjacent?
Arc Addition Postulate: The measure of the arc formed by 2 adjacent arcs is the sum of the measures of the 2 arcs.
On your handout m arc CDE = m arc CD + m arc DE.
Why does this have to be true?
If we are given the coordinates of the endpoints of a diameter of a circle how would we find the center?
How would we find the radius?
EX: The endpoints of the diameter of a circle are A(0,4) and B(-4,6)
Find the coordinates of the center and the length of the radius.
To find the coordinates of the center we will use the midpoint formula
M = [pic]
M = [pic]
M = (-2,5)
So the center of the circle is at (-2,5)
To find the length of the radius we use the distance formula and find the distance from the center to either of the points on the circumference. I will use the coordinates of the center and point A.
d = [pic]
d = [pic]
d = [pic]
d = [pic]
So the length of the radius is [pic].
EX The coordinates of the endpoints of the diameter of a circle are A(2,3), and B(-4,5). Find the coordinates of the center and the length of the radius.
Chapter 2-6 Congruent and Similar Figures
Congruent Figures
Congruent figures have exactly the same size and shape.
They may be facing in opposite directions, they may be different colors, but they have to be exactly the same size and shape.
Congruent Circles
Congruent circles have congruent radii.
Congruent Polygons
Congruent polygons have congruent corresponding parts. Matching vertices are corresponding vertices. All angles of one are the same as the angles of the other and every side of one has a corresponding equal side on the other.
Similar Figures
Two figures that have exactly the same shape, but not necessarily the same sizes are called similar figures.
Similar Circles
All circles are similar. They all have exactly the same shape!
Similar Polygons
Polygons are similar polygons if
1. Corresponding angles are congruent
2. Corresponding sides are proportional
Why do the angles have to be congruent?
Similarity Ratio
The ratio of the corresponding sides of 2 similar polygons is called the similarity ratio for those polygons.
Chapter 3-1 Reflections
Transformation - A transformation is a change in the position, shape or size of a figure. A transformation maps a figure onto its image.
Preimage - A preimage is the original figure before it is transformed.
Image - An image is the transformed figure. The image is usually identified by primes. If you are transforming triangle ABC, the image is usually called triangle A1B1C1
Isometry - An isometry is a transformation in which the original figure and its image are congruent.
Orientation - The orientation of an object is the way it is facing (a mirror reverses orientation). If whatever was on the left in the preimage is still on the left in the image and whatever was on the right in the preimage is still on the right in the image then the orientation has remained the same. If the left and right of the preimage have become reversed in the image then the orientation has been reversed.
The easiest way to tell if the orientation of a polygon has been reversed is to label the corresponding vertices of the image and the preimage. Then start at any vertex of the image and go clockwise around the figure reading the letters. Go to the image and start at the vertex corresponding to the vertex you started with on the preimage and go around clockwise. If the letters in the image are in the same order the orientation is the same. If the letters in the image are in the reverse order the image has been reversed.
EX:
[pic]
If figure 2 is the image of figure1, then the orientation of the figure has not changed. If I start at vertex A of figure 1 and go clockwise around the figure and read the vertices they are ABC. If I start at the image of A (A1) and go around figure 2 clockwise the image of B (B1) comes next and the image of C (C1) comes last. The vertices have remained in the same order.
[pic]
Now the orientation of the figures has been reversed. If I start at vertex A in figure 1 and go around clockwise the vertices are ABC. If I start at the image of A in figure 2 and go around the figure clockwise the vertices are A1C1B1. The corresponding vertices are no longer in the same order - they are in opposite order.
Reflection:
A reflection in line r is a transformation for which the following are true
• If a point A is on line r, then the image of A is itself (i.e.: A = A1)
• If a point B is not on r then r is the perpendicular bisector of BB1
A reflection reverses orientation. A reflection is an isometry.
EX: If you are standing in front of a mirror with your nose pressed up to the glass, then the image of the tip of your nose looks like it is pressed up to the glass too. So you could almost say that the image of the tip of your nose is the tip of your nose. If the mirror had no thickness, you wouldn’t be able to tell the image of the tip of your nose from the preimage.
Now let’s look at one of your shoulders. If you drew a line segment from the top of your shoulder to the top of the image of your shoulder, the part of the segment outside the mirror would appear to be the same length as the segment inside the mirror- the mirror bisects the line segment. Every point of your body appears to be exactly as far away from the outside of the mirror as the image appears to be on the inside.
EX: Triangle ABC has coordinates A(-3, 4), B(0,1), C(2,3)
Draw its reflection in a) the x axis b) the y axis
Chapter 3-2 Translations
A translation is a transformation that moves points the same distance and in the same direction.
• A translation is an isometry.
• A translation does not change orientation.
EX: An elevator – it moves people or objects all the same distance and in the same direction.
The distance and direction of a translation can be expressed as a vector. Vectors are line segments with an initial point and a terminal point. The diagram of a vector can look like a ray. Be sure you know what the difference is.
You can use ordered pair notation to represent vectors, . When this notation is used does not refer to a specific point on the coordinate axis. Instead x represents the horizontal change from the initial point to the terminal point and y represents the vertical change.
EX: If I am translating point A (3, 2) along the vector it goes to the point 4 units over on the x axis and 1 unit up on the y axis and is translated to point (7, 3).
EX: What is the image of P(3, 3) under the translation .
EX: What vector describes the translation from A(2,3) to B(-1,-3).
EX: What vector describes the translation from T(-3,-1) to P(3,2)
You can use matrices to translate figures in the coordinate plane. For example assume you have a triangle ∆DOG whose vertices are at D(-5, -3), O(2, 6), and G(5, -2) and you wanted to translate it using the translation vector . You could create a 2x3 matrix such that all the x-coordinates are in the first row and all the y-coordinates are in the second.
This is the matrix of the original triangle[pic]
Now create a matrix containing the translation vector that you will then add to the original matrix. Since matrix addition requires matrices of the same size, this matrix will also be 2x3, just repeat the vector as many times as necessary.
This is the matrix of the translation vector[pic]
Now use matrix addition to add the matrices[pic]
The coordinates of the translated image of ∆DOG, ∆D1O1G1 are D1(-9, 1), O1(-2, 10), and G1(1, 2).
Chapter 3-3 Rotations
A rotation turns a figure around a given point by a set number of degrees.
A rotation of x° about a point R is a transformation such that:
• For any point V, RV = RV1 and m TY. Find P on OY so that TY ( PY. Draw TP.
|Given: OY > TY and TY ( PY |[pic] |
| | |
|Prove: m(OTY > m(3 | |
|Statements |Reasons |
|1. YP ( YT |1. |
|2. m(1 = m(2 |2. |
|3. m(OTY = m(4 + m(2 |3. |
|4 m(OTY > m(2 |4. Comparison Property of Inequality |
|5. m(OTY > m(1 |5. |
|6. m(1 = m(3 + m(4 |6. |
|7. m(1 > m(3 |7. |
|8. m(OTY > m(3 |8. |
Theorem: If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.
Prove with an indirect proof.
|Properties of Inequality |
|For all real numbers a, b, c, and d: |
|Addition |If a > b and c ( d, then a + c > b + d |
|Multiplication |If a > b and c > 0, then ac > bc |
| |If a > b and c < 0, then ac < bc |
|Transitive |If a > b and b > c, then a > c |
|Comparison |If a = b + c and c > 0, then a > b |
Chapter 4-7 Bisectors and Locus
Definition: A locus is all the points that satisfy a given condition.
EX: As we learned before, the definition of a circle is: a circle is all the points in a plane at a given distance from a set point called the center. We therefore say that the locus of points in a plane at a set distance from a given point is a circle.
Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
This theorem follows from the definition of a perpendicular bisector and from the construction (remember the construction) of perpendicular bisectors.
Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
This theorem also follows from the definition and the construction.
These two theorems together show that the locus of points equidistant from the endpoints of a line segment are on the perpendicular bisector of that segment
EX: what is the locus of points in a plane 3 cm from a given line? What if it is not in a plane?
EX: what is the locus of points in a plane 3 cm from a given segment?
Definition: The distance from a point to a line is the length of the perpendicular segment from the point to the line.
The Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
Converse of the Angle Bisector Theorem: If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the angle bisector.
These theorems follow from the definition of an angle bisector.
These 2 theorems show that the locus of points in the interior of an angle that are equidistant from the sides of the angle are on the angle bisector.
Chapter 4-8 Concurrent Lines
Concurrent: When 3 or more lines intersect in a point they are concurrent. The point at which they intersect is the point of concurrency.
Review:
It is critical for you to remember what a perpendicular bisector and an angle bisector are, so if you don’t remember go look it up!
Theorem: The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.
Note- the perpendicular bisectors do not usually come from the vertices.
Theorem: The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.
Median of a Triangle: the median of a triangle is a segment whose endpoints are a vertex and the midpoint of the side opposite the vertex.
BE CAREFUL – a median is not the same thing as a perpendicular bisector. A median is like a perpendicular bisector because it bisects a side but a median comes from the opposite vertex and is not necessarily perpendicular to the side it bisects.
Altitude of a Triangle: the altitude of a triangle is a perpendicular segment from a vertex to the line containing the side opposite the vertex.
BE CAREFUL – an altitude is not necessarily a median because it does not necessarily bisect the opposite side. It is not necessarily the perpendicular bisector of the side because it always comes from the vertex.
There are 4 types of special lines associated with triangles:
Angle Bisectors
Perpendicular Bisectors
Medians
Altitudes
Theorem: The lines that contain the altitudes of a triangle are concurrent
Theorem: The medians of a triangle are concurrent
Chapter 5-1 Understanding Perimeter and Area
Perimeter of a polygon
The perimeter of a polygon is the sum of the lengths of its sides.
Examples:
▪ The perimeter of a square with side length ‘s’ is s + s + s + s or 4s.
▪ The perimeter of a rectangle with a side length ‘b’ and a side length ‘h’ is b + b + h + h or 2b + 2h
▪ You can use the distance formula to find the perimeter of any polygon in the coordinate plane
Area of a Polygon
The area of a polygon is the number of square units enclosed by the polygon. [pic]
Theorem: The area of a rectangle is the product of its base and its height.
For a rectangle of base b and height h, A = bh.
Postulate: The area of a square is the square of the length of a side.
For a square of side s, A = s2.
Postulate: If 2 figures are congruent their areas are equal.
Postulate: The area of a region is the sum of the areas of its nonoverlapping parts.
Chapter 5-2 Areas of Parallelograms and Triangles
Theorem: The area of a parallelogram is the product of any base and the corresponding height.
You can choose any side to be the base of the parallelogram.
An altitude is any line drawn from the side opposite the base perpendicular to the base.
The height is the length of the altitude.
The easiest way to prove this is geometrically – draw a parallelogram and label a base and height. Using the definition of a parallelogram and the definition of area show on your drawing how the area of the parallelogram can be deduced from the area of a rectangle.
Theorem: The area of a triangle is equal to half the product of any base and the corresponding height.
You can choose any side to be the base. The corresponding height is the length of an altitude drawn to the line containing that base.
The easiest way to prove this is geometrically – any triangle can be considered half of the appropriate parallelogram.
Chapter 5-3 The Pythagorean Theorem and Its Converse
Hypotenuse:
The hypotenuse in a right triangle is the side opposite the right angle. It is the longest side in the triangle.
Legs of a Right Triangle:
The sides of a right triangle other than the hypotenuse are called the legs of the right triangle.
Pythagorean Theorem: In a right triangle the sum of the squares of the lengths of the legs is equal to the square of the hypotenuse.
Pythagorean Triples:
Integers that could be the sides of a right triangle are called Pythagorean triples. They are integers such that the sum of the squares of two of them equals the square of the third.
EX: 3, 4, and 5 and 5, 12, and 13
Converse of the Pythagorean Theorem: If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
There are some very important corollaries to the Pythagorean Theorem and its converse, the triangle inequalities.
|Triangle Inequalities |
|In triangle ABC with longest side c … |
|If c2 = a2 + b2, then the triangle is a right |If c2 > a2 + b2, then the triangle is obtuse |If c2 < a2 + b2, then the triangle is acute |
|triangle | | |
|[pic] |[pic] |[pic] |
If you start with a right triangle with hypotenuse c (the side opposite the right angle C) and you make side c bigger, while you keep a and b the same size, the triangle becomes an obtuse triangle. If you make side c smaller the triangle becomes an acute triangle.
Chapter 5-4 Special Right Triangles
45(-45(-90( Triangles
Theorem: In a 45(-45(-90( triangle, both legs are congruent and the length of the hypotenuse is [pic] times the length of a leg.
30(-60(-90( Triangles
Theorem: In a 30(-60(-90( triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is [pic] times the length of the shorter leg.
Chapter 5-5 Areas of Trapezoids
Bases of a Trapezoid
The bases of a trapezoid are the parallel sides.
Legs of a Trapezoid
The legs of a trapezoid are the nonparallel sides.
Height of a Trapezoid
The height of a trapezoid is the perpendicular distance between the bases.
Theorem: The area of a trapezoid is half the product of the height and the sum of the parallel bases.
EX: Find the area of a trapezoid with bases 21 and 38 and height 16.
EX: Find the area of a trapezoid with one base 5 and one 7 and one base angle that is a right angle and one base angle that is 60°.
EX: What if we had that same trapezoid, except that the angle that was 60° is only 45°. Now what is the area?
Chapter 5-6 Areas of Regular Polygons
Regular Polygon:
A regular polygon is a polygon that is equilateral and equiangular. That means all the sides and all the angles are congruent.
You can circumscribe a circle around any regular polygon.
The center of a regular polygon is the center of the circumscribed circle.
The radius of a regular polygon is the distance from the center to any vertex.
The apothem of a regular polygon is the perpendicular distance from the center to a side.
Theorem: The area of a regular polygon is half the product of the apothem and the perimeter.
[pic]
Proof: You can divide any regular n-gon into n isosceles triangles. Let a represent the length of the apothem of the polygon. Let s represent the length of each side. Then a is the altitude to the base, s, of each of the triangles.
The area of each of the triangles is
The area of the polygon is n times the area of each triangle (there are n triangles). So the area of the polygon is
But is the perimeter of the polygon. So the area of the polygon can be written
EX: Find the area of a decagon with a 12in apothem and 8in sides.
EX: Find the area of an equilateral triangle with a radius of 14.
EX: Try another one – find the area of a square with radius 4.
Chapter 5-7 Circles: Circumference and Arc Length
Circumference of a Circle:
The circumference of a circle is the distance around the circle.
C = πd C = 2πr (because d = 2r)
Where d stands for the length of a diameter of the circle and r stands for the length of a radius.
Concentric Circles:
Circles that lie in the same plane and have the same center are called concentric circles.
Arc Length:
The length of arc AB is the length of the piece of the circumference between A and B.
The measure of an arc in degrees tells what fraction of the circumference that arc is.
If an arc is 90° it represents ¼ of the way around the circle, because [pic]. So the length of that arc is ¼ of the length of the circumference.
If an arc is 60°, it represents [pic] of the way around the circle, because [pic].
So the length of that arc is [pic] of the length of the circumference.
If an arc is m°, it represents [pic] of the way around the circle.
So the length of that arc is [pic] of the length of the circumference.
The formula for arc length:
The length of an arc of a circle is the product of the ratio [pic] and the circumference of the circle.
Given circle O, with radius r, and arc AB
Length of AB = [pic]
Congruent Arcs:
Arcs that have the same measure and are in the same circle or congruent circles.
Chapter 5-8 Areas of Circles, Sectors and Segments of Circles
Area of a Circle:
The area of a circle is the product of ( and the square of the radius, A = (r2.
Sector of a Circle:
A sector of a circle is the region bounded by 2 radii and their intercepted arc.
A piece of pizza is a sector of a circle.
You name a sector by using one endpoint of the arc, the center of the circle and the other endpoint of the arc.
If the arc that bounds a sector is half a circle (1800), then the area of the sector will be half the area of the circle.
If the arc that bounds a sector is 1/3 of a circle (1200), then the area of the sector will be 1/3 the area of the circle.
Area of a sector:
The area of a sector of a circle is the product of the ratio [pic] and the area of the circle
Area of sector AOB = [pic]
EX: Find the area of a sector of a circle bounded by an arc of 600, if the circle has a diameter of 26.
Segment of a Circle:
The part of a circle bounded by an arc and the segment joining its endpoints is a segment of a circle.
To find the area of a segment of a circle you have to find the area of the circle, then the area of the sector and then subtract the area of the triangle.
Chapter 6-1 Space Figures and Nets
Polyhedron – a 3 dimensional figure whose surfaces are polygons.
Faces – the polygons that are the surfaces.
Edge – intersection of 2 surfaces.
Vertex – point where edges intersect.
Net – a 2-dimensional pattern that you can fold to form a 3-dimensional figure.
Ex: Any box is a polyhedron. The sides are faces. Boxes are made from nets (flat pieces of cardboard that are folded to create the boxes).
Euler’s Formula – the number of edges of a polyhedron can be calculated as [pic].
(v = # of vertices, f = # of faces, and e = number of edges)
Chapter 6-2 Surface Areas of Prisms and Cylinders
Prism – a polyhedron with 2 congruent parallel bases.
A prism is named for the shape of its bases.
Lateral faces – the faces of a prism other than the bases.
Altitude of a prism – the perpendicular segment that joins the planes of the bases.
Height of a prism – the length of an altitude.
Right prism – the lateral faces are rectangles and every lateral edge is an altitude.
Oblique prism – any prism that is not a right prism.
You can assume a prism is a right prism unless you are told otherwise.
Lateral area of a prism – sum of the areas of the lateral faces.
Surface area of a prism – sum of the lateral area and the area of the 2 bases.
Your book uses the word Surface Area. The makers of the EOC call it Total Area. They mean exactly the same thing.
You book uses L.A. as an abbreviation for lateral area and the EOC uses just L.
Your book uses S.A. to stand for surface area and the EOC uses T to stand for total area and they are both the same.
Theorem: The lateral area of a right prism is the product of the perimeter of the base and the height. L. A. = ph or L = ph.
Theorem: The surface area of a right prism is the sum of the lateral area and the area of the 2 bases. S.A. = L. A. + 2B or T = L + 2B.
Cylinder – a 3-dimensional figure with 2 parallel, congruent circles as bases.
Altitude – the perpendicular segment that joins the plane of the bases.
Height – the length of the altitude.
Lateral Area – the product of the circumference of the base and the height of the cylinder.
L. A. = 2(rh or L = 2(rh.
The surface area of a cylinder is the sum of the lateral area and the area of the bases
S. A. = L. A. + 2B or T = 2(r(h + r).
Example: A roll of paper towels is cylinder
Chapter 6-3 Surface Areas of Pyramids and Cones
Pyramid – a polyhedron in which one face (the base) can be any polygon and the other faces (lateral faces) are triangles that meet at a common vertex called the vertex of the pyramid.
A pyramid is named by the shape of the base.
Altitude – the perpendicular segment from the vertex to the plane of the base.
Height – the length of the altitude.
Slant height (l) – the length of an altitude of a lateral face.
Example: The Egyptian pyramids.
Regular pyramid – the base is a regular polygon and the lateral faces are congruent triangles.
Lateral Area:
L. A. (Lateral Area) = ½ lp, or L = ½ lp, where p is the perimeter of the base.
S. A. (Surface Area) = L. A. + B or T (Total Area) = L + B
[pic]a is the height and b is the slant height
Cone – Like a pyramid but the base is a circle.
Altitude – the perpendicular segment from the vertex to the center of the base.
Height – length of the altitude.
Slant height (l) –distance from the vertex to a point on the edge of the base.
Lateral Area of a cone is one-half the circumference of the base times the slant height.
( [pic] ) which can be simplified to: L. A. = (rl or L = (rl
S. A. = L. A. + B or T = (r(l + r)
Example: An ice cream cone is a cone.
Hint: to find l if you know h and r or to find h if you know l and r or to find r if you know l and h just use the Pythagorean theorem.
Very Important Point ( The height of a regular pyramid or a cone will connect the vertex to the center of the base. For example, use this diagram. Because the height of 12cm connects the vertex to the center of the base, the length of the darker segment is the apothem of the base, or 9cm in the instance (recall how to find the apothem in a regular polygon). So to find the slant height, I have a right triangle with legs of 9 and 12 and can use Pythagoras to find the hypotenuse which is the slant height.
Chapter 6-4 Volumes of Prisms and Cylinders
Volume – the space a figure occupies, measured in cubic units.
Volume of prisms
V = Bh Where B = area of base and H = length of altitude.
Volume of cylinders
V = Bh = (r2h
Chapter 6-5 Volumes of Pyramids and Cones
Volume of pyramids
V = [pic]Bh Where B = area of base and h = length of perpendicular segment from the vertex to the plane of base.
Volume of cones
V = [pic]Bh = [pic](r2h
Where B = the area of the base and h = the length of the altitude and r = the radius of the base.
Chapter 6-6 Surface Areas and Volumes of Spheres
A sphere is the locus of points in space equidistant from a given point, called the center.
A radius of a sphere is the distance from the center of the sphere to the outside of the sphere.
A diameter of a sphere is a segment passing through the center of the sphere, with endpoints on the surface of the sphere
V = [pic](r3
S. A. = 4(r2 or A = 4(r2
Chapter 6-7 Composite Space Figures
A composite space figure combines 2 or more 3-dimensional figures. The surface area and volume of a composite space figure is the sum of the surface areas and volumes, respectively, of the figures that are combined.
Be careful when calculating composite space figures. Very often parts of one figure are covered by another or perhaps only part of a figure is included in the final figure.
[pic]For example, this figure is a cone and half of a sphere.
Chapter 6-8 Geometric Probability
Geometric probability uses geometric figures to represent occurrences of events.
The probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes.
If P represents the probability of an event then
P = [pic]
Geometric probability compares occurrences by comparing measurements of geometric figures.
Chapter 7-1 Parallel Lines and Related Angles
Angles Formed by Intersecting Lines
Transversal - a line that intersects 2 coplanar lines at a distinct point on each line.
A transversal, t, creates several different kinds of angles with the lines l and m that it intersects
Corresponding angles: the angles that lie on the same side of the transversal t and in corresponding positions relative to l and m.
Same Side Interior Angles: angles that are on the same side of the transversal and are between the lines that it intersects.
Alternate Interior Angles: angles that are on opposite sides of the transversal t (alternate sides) and are between the lines that it intersects (interior to them - inside)
[pic]
Same Side Exterior Angles: angles that are on the same side of the transversal and are outside the lines that it intersects.
Alternate Exterior Angles: angles that are on opposite sides of the transversal (alternate sides) and are outside the lines that it intersects (exterior to them - outside)
Postulate 7-1 - If two parallel lines are cut by a transversal all the corresponding angles are congruent.
[pic]
Theorem 7-1: If 2 parallel lines are cut by a transversal alternate interior angles are congruent.
Proof of Theorem 7-1
Given: m ║ l
Prove: (4 ( (6
|Statements |Reasons |
|1. m ║ l |1. Given |
|2. (2 ( (6 |2. If ║ lines, then corresponding angles are congruent. |
|3. (2 ( (4 |3. Vertical angles are congruent. |
|4. (4 ( (6 |4. Transitive Property of Congruence. |
Theorem 7-2: If 2 parallel lines are cut by a transversal, then pairs of the same side interior angles are supplementary.
Proof of Theorem 7-2
Given: m ║ l
Prove: (4 and (5 are supplementary
|Statements |Reasons |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
Be careful: If two lines are cut by a transversal, the original lines and the transversal will form corresponding angles, alternate interior angles and same side interior angles. However, if the lines are not parallel the corresponding angles and the alternate interior angles will not be congruent and the same side interior angles will not be supplementary.
Chapter 7-2 Proving Lines Parallel
Postulate 7-2: If two lines are cut by a transversal so that a pair of corresponding angles are congruent, then the lines are parallel.
Theorem: If two lines are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel.
Theorem: If two lines are cut by a transversal so that a pair of same side interior angles is supplementary, then the lines are parallel.
Chapter 8-1 Proving Triangles Congruent: SSS and SAS
Review – what does it mean to say 2 figures are congruent? They are exactly the same size and shape. All of the angles have the same measure and all of the sides are the same size.
For triangles that means that if all 3 angles of one triangle are congruent to all three angles of another triangle and if all 3 sides of the first triangle are congruent to all 3 sides of the second triangle then the triangles are congruent.
So if we have 2 triangles that are congruent we can make 6 congruence statements: 3 about angles and 3 about sides.
SSS Postulate: Side – Side - Side Postulate
If 3 sides of one triangle are congruent to 3 sides of another triangle, then the 2 triangles are congruent.
SAS Postulate: Side – Angle – Side Postulate
If 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of another triangle, then the 2 triangles are congruent.
Chapter 8-2 Proving Triangles Congruent: ASA and AAS
ASA Postulate: Angle – Side Angle Postulate
If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of another triangle, then the 2 triangles are congruent.
AAS Theorem: Angle – Angle – Side Theorem
If 2 angles and a non included side of one triangle are congruent to 2 angles and the corresponding non included side of another triangle, then the triangles are congruent.
Chapter 8-3 Congruent Right Triangles
HL Theorem: Hypotenuse Leg Theorem
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
Chapter 8-4 Using Congruent Triangles in Proofs
CPCTC: Corresponding parts of congruent triangles are congruent.
We will use the fact that corresponding parts of congruent triangles are congruent so often in proofs that we use the initials CPCTC as the reason to conclude 2 parts of congruent triangles are congruent in proofs.
Chapter 8-5 Using More Than One Pair of Congruent Triangles
There are no formulas to memorize here or some set of instructions that will always work. The only way to get good at this is to do lots of problems. However, there are a few hints and tips.
Tips
If you are trying to do a proof that involves overlapping triangles, draw a separate diagram in which you draw them separately.
On your new diagram immediately mark all the corresponding sides and angles that you know.
Remember that any side or angle that was a shared side or angle in the original diagram is a congruent corresponding side of the 2 triangles in the new diagram.
If you are trying to prove 2 smaller triangles that are parts of bigger triangles congruent, sometimes you should prove the bigger triangles congruent first and then use corresponding parts from the bigger triangles to prove the smaller triangles congruent.
Chapter 9-1 Properties of Parallelograms
Definition: A quadrilateral is a parallelogram if both pairs of opposite sides are parallel.
Theorem: The opposite sides of a parallelogram are congruent.
Theorem: The opposite angles of a parallelogram are congruent.
Theorem: The diagonals of a parallelogram bisect each other.
Theorem: If 3 or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.
Chapter 9-2 Proving That a Quadrilateral is a Parallelogram
Theorem: If the diagonals of a quadrilateral bisect each other the quadrilateral is a parallelogram
Theorem: If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram
Theorem: If both pairs of opposite sides of a quadrilateral are congruent then the quadrilateral is a parallelogram
Theorem: If both pairs of opposite angles of a quadrilateral are congruent then the quadrilateral is a parallelogram
Chapter 9-3 Properties of Special Parallelograms
Rhombus:
Definition: A rhombus is a parallelogram in which all 4 sides are congruent
Theorem: Each diagonal of a rhombus bisects the 2 angles of the rhombus that it cuts
Theorem: The diagonals of a rhombus are perpendicular to each other
Theorem: The area of a rhombus is half the product of its diagonals
A = ½ d1d2
(of course the formula A = bh still works, since the rhombus is a parallelogram- the formula you use depends on what information you have)
I can use the formula A = ½d1d2 to find the area of a square.
Theorem: If one diagonal of a parallelogram bisects 2 angles of the parallelogram, then the parallelogram is a rhombus
Theorem: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus
Be careful with both of these. The quadrilateral has to be a parallelogram in the first place. If it isn’t a parallelogram, it can’t be a rhombus
Rectangles:
Definition: A rectangle is a parallelogram in which all 4 angles are right angles.
Theorem: The diagonals of a rectangle are congruent
Theorem: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
If the diagonals of a parallelogram are both congruent and perpendicular, what kind of quadrilateral is it?
This brings up a very important relationship.
A rhombus is a parallelogram.
A rectangle is a parallelogram.
A rhombus is not usually a rectangle.
A rectangle is not usually a rhombus.
When a rhombus is a rectangle, it is a square.
When a rectangle is a rhombus, it is a square.
Chapter 9-4 Trapezoids and Kites
Trapezoids
Definition: A trapezoid is a quadrilateral in which 2 opposite sides are parallel and the other 2 opposite sides are not parallel. The parallel sides are called the bases and the other sides are called the legs.
Definition: Each pair of angles adjacent to the base of a trapezoid is called the base angles of the trapezoid.
Definition: An isosceles trapezoid is a trapezoid in which the 2 legs are congruent
Theorem: The base angles of an isosceles trapezoid are congruent
Theorem: The diagonals of an isosceles trapezoid are congruent
EX: Given isosceles trapezoid FLAG with LF and AG as the congruent legs.
LA = 15, FG = 25, LF = AG = 13.
Find the area of the trapezoid.
You need the formula for the area of an isosceles trapezoid and you need to find the height. The formula is on your formula sheet. Where does it come from?
Kites
Definition: A kite is a quadrilateral with 2 pairs of adjacent sides congruent and no opposite sides congruent.
Theorem: The diagonals of a kite are perpendicular.
Do the diagonals of a kite bisect each other?
Does one bisect the other?
What other quadrilateral has diagonals that are perpendicular?
EX: Given kite RSTW with RS (RW and ST ( WT.
If RS = 15, ST = 13, and SW = 24, find RT.
Which angles in a kite are congruent? Are both pairs of opposite angles congruent, either one, neither?
PROPERTIES OF QUADRILATERALS
| |Parallelogram |Rhombus |Rectangle |Square |Trapezoid |Isosceles |Kite |
| | | | | | |Trapezoid | |
|All sides ( | | | | | | | |
|Opposite sides ( | | | | | | | |
|1 pair opposite | | | | | | | |
|sides ( | | | | | | | |
|Opposite sides | | | | | | | |
|parallel[pic] | | | | | | | |
|1 pair opposite sides | | | | | | | |
|parallel | | | | | | | |
|Opposite angles ( | | | | | | | |
|1 pair of opposite | | | | | | | |
|angles ( | | | | | | | |
|Adjacent angles ( | | | | | | | |
|All angles, are right | | | | | | | |
|angles | | | | | | | |
|Diagonals bisect each | | | | | | | |
|other | | | | | | | |
|Diagonals are | | | | | | | |
|perpendicular | | | | | | | |
|Each diagonal bisects | | | | | | | |
|the opposite angles | | | | | | | |
|Diagonals are ( | | | | | | | |
Chapter 10-1 Ratio, Proportion, and Similarity
Definition: 2 polygons are SIMILAR if
1) Corresponding angles are congruent
2) Corresponding sides are proportional
Definition: Similarity ratio
The ratio of the lengths of corresponding sides of similar figures
Write as a:b or [pic]
Definition: Proportion
A proportion is the statement that 2 ratios are equal
[pic] = [pic] or a:b = c:d
An extended proportion is a proportion of the form
[pic] = [pic] = [pic] or a:b = c:d = e:f
Properties of proportions:
[pic] = [pic] is equivalent to
1) ad = bc Cross Product Property
2) [pic] = [pic]
3) [pic] = [pic]
4) [pic] = [pic]
We can use these different properties to solve proportions
EX: [pic]
25x = 10(15)
x = [pic]
x = 6
EX: [pic]
4(x + 3) = 3(14)
4x + 12 = 42
4x = 30
x = 7.5
A scale drawing compares dimensions in different units. The scale compares each measurement in the drawing to the actual measurement being represented. An example is the scale on a street map where one inch on the map might represent an actual distance of 150 miles.
Chapter 10-2 Proving Triangles Similar: AA, SAS, and SSS
Theorem: AA Similarity Theorem
If 2 angles of one triangle are congruent to 2 angles of another triangle the triangles are similar
Theorem: SAS Similarity Theorem
If an angle of one triangle is congruent to an angle of another triangle and the sides including the 2 angles are proportional, then the triangles are similar
Theorem: SSS Similarity Theorem
If the corresponding sides of 2 triangles are proportional then the triangles are similar
What would the similarity ratio have to be in the SAS theorem and the SSS theorem to make the triangles congruent?
Chapter 10-3 Similarity in Right Triangles
You need to understand the concept of the geometric mean to understand this section.
Definition: Geometric Mean
The geometric mean of a and b is the positive number x such that
[pic] = [pic]
EX: To find the geometric mean of 3 and 12.
If x is the geometric mean then
[pic] = [pic] so
x2 = 36
x = 6
To find the geometric mean of 4 and 5
If x is the geometric mean
[pic] = [pic]
x2 = 20
x = [pic] = 2[pic]
So what does the geometric mean mean anyway?
Theorem: The altitude to the hypotenuse of a right triangle divides the triangle into 2 right triangles that are similar to the original triangle and to each other
Corollary 1: The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.
Corollary 2: The altitude to the hypotenuse of a right triangle intersects it so that the length of each leg is the geometric mean of the length of its adjacent segment of the hypotenuse and the length of the entire hypotenuse.
|(a, b, and c are the sides of a right triangle with c the hypotenuse. |[pic] |
|d is an altitude to the hypotenuse and e and f are the segments the | |
|hypotenuse is divided into by that altitude.) | |
|Corollary 1 | |
|[pic] | |
|Corollary 2 | |
|[pic] and [pic] | |
Chapter 10-4 Proportions and Similar Triangles
Side Splitter Theorem:
If a line is parallel to one side of a triangle and intersects the other 2 sides, then it divides those sides proportionately.
Corollary: If 3 parallel lines intersect 2 transversals, then the segments intercepted on the transversals are proportional.
Triangle Angle Bisector Theorem:
If a ray bisects an angle of a triangle, then it divides the opposite side into 2 segments that are proportional to the two other sides of the triangle.
Chapter 10-5 Perimeters and Areas of Similar Figures
Theorem: If the similarity ratio of 2 similar plane figures is a : b, then
1) the ratio of their perimeters is a : b.
2) the ratio of their areas is a2 : b2.
Chapter 10-6 Areas and Volumes of Similar Solids
Theorem: If the similarity ratio of 2 similar solids is a :b, then
1) the ratio of their corresponding areas is a2 : b2.
2) the ratio of their volumes is a3 : b3.
Chapter 11-1 The Tangent Ratio & Chapter 11-2 The Sine and Cosine Ratios
SOH-CAH-TOA
SOH-CAH-TOA is the magic word you need to remember to unlock the mysteries of right triangle trigonometry.
If you know the length of one side of a right triangle and the measure of one of the non-right angles you can find the lengths of all 3 sides. If you know the lengths of 2 sides you can find the length of the third side (you already know how to do that – the Pythagorean Theorem) and the measure of all of the angles.
You can do that by means of special proportions that apply to right triangles. These only apply to right triangles.
These proportions are called the sine, cosine and tangent ratios.
If θ (this is the Greek letter theta)is one of the non-right angles in a right triangle and opposite means the leg opposite θ and adjacent means the leg next to θ, (note: legs not the hypotenuse) then:
Sine θ = [pic]
Cosine θ = [pic]
Tangent θ = [pic]
That’s what the mnemonic will help you to remember
SOH – sine is opposite over hypotenuse
CAH – cosine is adjacent over hypotenuse
TOA – tangent is opposite over adjacent
There are common abbreviations that are always used
Sine ( is sin (
Cosine ( is cos (
Tangent ( is tan (
|[pic] |The given triangle is a right triangle. Find the sin θ, |
| |cos θ and tan θ. |
| | |
| |Sin θ = |
| | |
| |Cos θ = |
| | |
| |Tan θ = |
All of these ratios are true because all right triangles that have one of their non-right angles congruent are similar. So if you know you have a right triangle with an angle of 36°, it is similar to every right triangle with an angle of 36°. Be sure you understand why this is true and can explain it.
It doesn’t matter how big or small either of the triangles are, their sides will be in the same proportions. That means once you know that proportion it is true for every right triangle that has angles the same size.
Trig ratios are probably the single most important use of similar triangles.
You will have to have your calculators for this chapter and for tests on this chapter.
What is the sin of 30(? What is the cos of 30( in radical form?
What is the tan of 30(?
What is the sin and the cos of 45( - why is this one question?
What is the tan of 45(?
Inverse Trig Functions
The inverse trig functions give us the angle if we know the value of the trig function.
Arcsin M means the angle whose sine is M. We write sin-1M.
Arccos N means the angle whose cosine is N. We write cos-1N.
Arctan L means the angle whose tangent is L. We write tan-1L.
Remember the inverse trig function is an angle!
Let’s look at a triangle and see how this works:
|[pic] |We are given that angle C is a right angle. Side AC = 8 and side BC = 15. |
| |We want to “solve the triangle”. That means we want to find every side and angle that was not given. |
| |How will we find side BA? |
Once we know all three sides we can use the inverse of any of the trig functions to find the non right angles.
For example we can use sine to find angle A
Sin A = [pic] = .88
So A is the angle whose sin is .88
We write that as arcsin A = sin-1A = .88
Put that in your calculator as 2nd sin .88 and you will find A = 61.93
Find A using cos and tan.
Now find angle B?
Try a few:
EX 1: Find the angles if
Tan A = 0.5 Cos B = [pic] Sin C = .76
EX 2: Given that the tan of A = 1. Find A without using your calculator or inverse trig functions.
EX 3: What is cos(sin-1[pic] )?
The inverse of sin A, is sin-1A.
So what is sin(sin-1.5)?
What is the sin-1(sin 30()?
Chapter 11-3 Angles of Elevation and Depression
What use is Trigonometry anyway?
Surveyors use special instruments called the transit and the theodolite. The surveyor sets the a horizon line on the instruments perpendicular to the direction of gravity and then sights up at the top of a mountain or whatever the surveyor needs to find the height of. The instrument tells the surveyor exactly what the angle is to the top of whatever it is pointed at. The surveyor then measures the distance of the instrument from the base of the object. Now the surveyor has a right triangle to work with. The height of the object is the side of the right triangle opposite the angle and the distance of the object from the instrument is the side of the triangle adjacent to the angle. By using the tangent of the angle the surveyor can determine the height of the object. This can be very important in making maps.
Using the cosine the surveyor can determine the length of the hypotenuse. This can be very important in determining the amount of material needed to build a ramp or a road.
The angle from the horizon to the top of an object is called the angle of elevation.
Air traffic controllers use trig to tell pilots what their angles of descent should be or how far from the runway they should begin their descent to make sure the planes end up on the runway. If a plane is about to start a descent and it is a certain distance from the airport and flying at a certain height, the tangent will tell the air traffic controllers what angle the plane should come in at. If the air traffic controllers want the plane to come in at a certain angle because of other traffic they can tell the pilot at what distance from the airport to begin the descent. If a line is drawn straight ahead from the plane, the angle between that line and the line the plane is going to come down on is called the angle of descent, or the angle of depression. That angle is equal to the angle of elevation, which is the angle between the ground and the line that represents the path the plane will come down on.
|1. You stand 40 ft from a tree. The angle of elevation from you to the tree is 47°. How tall is the tree? |
| |
| |
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|2. The angle of elevation to a building in the distance is 22°. You know that the building is approximately 450 ft tall. Estimate the distance to |
|the building. |
| |
| |
| |
| |
|3. An airplane is flying at an altitude of 10,000 ft. The airport at which it is scheduled to land is 50 miles away. Find the angle at which the |
|airplane must descend for landing. (There are 5280 feet in one mile.) |
| |
| |
| |
|4. A lake measures 600 ft across. A lodge stands on one shore. From your point on the opposite shore, the angle of elevation to the top is 4°. |
|How tall is the lodge? |
| |
| |
| |
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|5. A church wishes to build an access ramp for wheelchairs. The main hall of the church is 8 ft above sidewalk level. If the architect recommends |
|a grade (angle of elevation) of 6°, how long must the access ramp be? |
| |
| |
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|6. Two buildings stand 90 ft apart at their closest points. At those points, the angle of depression from the taller building to the shorter |
|building is 12°. How much taller is the taller building? |
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Chapter 11-4 Vectors and Trigonometry
A vector is a quantity that has magnitude (size) and direction.
To be a vector it must have both.
EX: I walk 2 miles a day does not describe a vector. I told you the magnitude of my walk but not the direction.
I walk east until I reach the river does not describe a vector. I told you the direction but not the size.
I walk east for 1 mile does describe a vector. It gives you direction and magnitude.
We used ordered pair vectors to describe translations. When we said we moved point (3,2) along the vector we said how far we moved it and in what direction. We moved it to the left 4 and up 3, to end up at the point (-1, 5).
Vectors can be written as ordered pairs or they can be drawn and they look like segments of rays. They look like rays but they have not only a specific end point on one side, but also a specific stopping point on the other side. The length of the ray segment gives its magnitude and there is a dot on the starting point and an arrow at the ending point to show the direction.
Often the direction of a vector is given in relation to a coordinate axis with an angle. For example when we say a wind is coming from the northwest we mean we have an imaginary coordinate axis and the wind is coming from 45( north of west.
We can use Pythagoras and trigonometry to calculate the magnitude and direction, respectively, of a vector as in this example:
Chapter 11-5 Adding Vectors
Vectors are often designated by a single lowercase letter in bold, such as u.
The sum of two vectors is called the resultant.
You can find the resultant of 2 vectors graphically. That is called the head to toe method.
Why do we call this the head to toe method?
You can also add two vectors analytically.
To add 2 vectors a and b, express the vectors in ordered pair notation and then add the x coordinates of each and then the y coordinates of each to get the resultant vector, which is the sum of the vectors.
If a = and b = , then to find the resultant vector c : c =
You can add vectors in any order: a + b = b + a
Chapter 11-6 Trigonometry and Area
In chapter 5 you learned that the area of a regular polygon is [pic], where a is the apothem and p is the perimeter.
What is the apothem?
Draw a regular polygon here and label the apothem and the radius.
You could only use that formula if you knew the length of the apothem or if the central angle formed by the radii of the polygon was 60°, 90° or 120°. Now you can use that formula for any regular polygon as long as you know the length of one side because you can use trig to find the length of the apothem.
The area of a triangle is A= ½bh. But what if you only know the length of 2 sides of the triangle and the measure of the angle between them? You can use trig to find the length of the altitude.
If you know the lengths of sides b and c of a triangle and the measure of angle A, (the included angle) the formula for the area of the triangle is
Area = ½bc(sinA)
Chapter 12-1 Circles in the Coordinate Plane
The equation of a circle
The standard form of the equation of a circle with center (h, k) and radius r is:
[pic]
For any point designated the center of a circle, use constants (h, k) as the ordered pair for the coordinates of the center.
Use (x, y) for the coordinates of any point on the circle.
You can now use the distance formula to find the radius.
Remember the distance formula …. [pic]
If the center of a circle is (h, k) and a point on the circle is (x, y), then the radius is
[pic] and if you square both sides, you get the standard form of the equation of a circle [pic]
Write equation of circle with center (5, -2) and radius 7.
[pic] (substitute into standard form)
[pic] (simplify the equation)
You can find the equation of a circle if you know the center and any point on the circle.
Any point on the circle is equidistant from the center and the distance is the radius. Remember the definition of a circle. So if you find the distance you have the radius.
You can find the equation of a circle if you know the endpoints of any diameter.
The midpoint of the diameter is the center and you can find the distance from the center to either endpoint to get the length of the radius (or you can find the length of the diameter and take half of it).
Chapter 12-2 Properties of Tangents
Definition: A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point.
Definition: The point of tangency is the point where the circle and the tangent intersect
Theorem: If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
Theorem: If a line in the same plane as a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.
Theorem: Two segments tangent to a circle from a point outside the circle are congruent.
[pic]AB ( AC
Definition: If each side of a triangle is tangent to a circle at one point then the triangle is circumscribed about the circle and the circle is inscribed in the triangle.
[pic]
Chapter 12-3 Properties of Chords and Arcs
Definition: A chord is a line segment in a circle with endpoints on the circumference of the circle.
[pic]Is a diameter a chord? A radius?
Prior Knowledge:
A central angle is an angle in a circle formed by 2 radii (the vertex is at the center of the circle).
An arc of a circle is a portion of the circumference.
The measure of an arc of a circle is the same as the measure of the central angle formed by drawing the radii from the endpoints of the arc to the center of the circle (we say the measure of an arc is the measure of its central angle).
[pic]
Theorem: In the same circle, or in congruent circles:
▪ Congruent central angles have congruent arcs
▪ Congruent arcs have congruent central angles
Theorem: In the same circle or congruent circles:
▪ Congruent chords have congruent arcs
▪ Congruent arcs have congruent chords
Theorem: A diameter that is perpendicular to a chord bisects the chord and its arc.
[pic]What about a radius? How about any part of a diameter?
Theorem: The perpendicular bisector of a chord contains the center of the circle.
Theorem: In the same circle or in congruent circles:
▪ Chords equidistant from the center are congruent
▪ Congruent chords are equidistant from the center
Chapter 12-4 Inscribed Angles
Definition: An angle in a circle is said to be an inscribed angle if the vertex of the angle is on the circumference of the circle and the sides of the angle are chords of the circle.
Definition: The intercepted arc of an inscribed angle is the arc formed on the circumference of the circle by the chords that are the sides of the angle.
Definition: A polygon is inscribed in a circle if all of its vertices lie on the circle. The circle is said to be circumscribed about the polygon.
Theorem: The measure of an inscribed angle is half the measure of its intercepted arc.
Corollary 1: Two inscribed angles that intercept the same arc are congruent
Corollary 2: An angle inscribed in a semicircle is a right angle.
Corollary 3: The opposite angles of a quadrilateral inscribed in a circle are supplementary.
Theorem: The measure of an angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc.
[pic]
Chapter 12-5 Angles Formed by Chords, Secants, and Tangents
Definition: A secant is a line that intersects a circle in 2 points. Secants always contain chords.
Theorem: The measure of an angle formed by two chords that intersect inside a circle is half the sum of the measures of the intercepted arcs.
Theorem: The measure of an angle formed by 2 secants, two tangents, or a secant and a tangent drawn from a point outside the circle is half the difference of the measures of the intercepted arcs.
Chapter 12-6 Circles and Lengths of Segments
Theorem: If 2 chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the other chord.
Theorem: If two secant segments are drawn from a point outside a circle the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.
Theorem: If a tangent and a secant are drawn from a point outside the circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.
-----------------------
Quadrilateral
2 Pair of
Parallel Sides
1 Pair of
Parallel Sides
No Pairs of
Parallel Sides
A
C
B
A1
C1
B1
Figure 1
Figure 2
A
C
B
A1
C1
B1
Figure 1
Figure 2
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
l
m
................
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