Trigonometry
Geometry Lesson Notes 1.3A Date ________________
Objective: Find the distance between two points.
Distance between two points: the measure of the line segment between the points.
When two points are on a number line:
To find AB, the measure of [pic], with coordinates a and b on the number line, take the difference of the values.
d = a – b or b – a
The measure of a segment is always positive, so take the absolute value of the difference.
(Note: The absolute value of the difference is not always the same as the difference of the absolute values.)
Example 1 (p 21): Find Distance on a Number Line
Find AB for different values of A and B.
Find the distance between
A(6, 3) and B(6, (5).
Find CD, given C((4, 7) and D(1, 7).
Find the measure of [pic].
(Recall the Pythagorean Theorem.)
For two points on the coordinate plane:
Distance Formula: The distance, d, between two points with coordinates
(x1, y1) and (x2, y2) is given by the formula:
[pic]
Example 2 (p 21): Find Distance on a Coordinate Plane
Find ST, the length of the segment with endpoints S((3, 4) and T(6, 1).
( HW: A6 pp 25-27 #13-18, 19-29 odd, 30
Geometry Lesson Notes 1.3B Date ________________
Objective: Find the midpoint of a segment.
Midpoint of a segment: the midpoint, M, of [pic] is the point between P and Q such that
PM = MQ.
Note: If M is between P and Q, then P and M and Q must be collinear.
On a number line,
The coordinate of the midpoint of the line segment whose endpoints have
the coordinates a and b is given by the formula:
[pic]
Example 3a (p23): Find Coordinates of the Midpoint
Try it on the number line below.
[pic]
Find point b on the number line if the midpoint of the segment is 15 and the value of
point a is (20.
Plot the points A(4, (3) and B(8, (3).
Draw [pic].
What is the midpoint of the line
segment?
Plot the points C(0, 8) and D((7, 8).
Draw[pic].
What is the midpoint of the line
segment?
Draw [pic].
What is the midpoint of the line segment?
On the coordinate plane,
The coordinates of the midpoint, M, of a line segment whose endpoints
have the coordinates (x1, y1) and (x2, y2) are given by the formula:
[pic]
Example 3b (p23): Find Coordinates of the Midpoint:
Find the midpoint of [pic] for A(4, (3) and D((7, 8).
Find the coordinates of the midpoint of [pic] for G(8, −6) and H(−14, 12).
Example 4 (p 23): Find the coordinates of the endpoint!
Find the coordinates of D if E(−6, 4) is the midpoint of [pic] and F has
coordinates (−5, −3).
NOTE: THIS IS AN EXAMPLE OF USING A FORMULA IN REVERSE!
IMPLICATION: What does it mean to be a midpoint? What must be true?
The midpoint of a segment splits the segment into two congruent segments.
What is the implication if two segments are congruent?
NOTE: YOU WILL USE IMPLICATIONS TO WRITE EQUATIONS TO DESCRIBE
GEOMETRIC RELATIONSHIPS!
Example 5 (p 23): Use Algebra to Find Measures
What is the measure of [pic] if Q is the midpoint of [pic]?
Use the implication of what it means to be a midpoint to solve the problem!
A ½ B 4 C 4½ D 9
Practice:
Given S((3, 4) and T(6, 1). Find the midpoint of [pic].
* Given M((30, 2) is the midpoint of [pic] and K(12, 11), find J.
* Given S is the midpoint of [pic], [pic], and [pic].
What does it mean to say that S is the midpoint of [pic]? What must
be true? Draw a sketch.
Find x and the measure of [pic].
* Given U is between T and V, [pic], TU = 9, and [pic].
Find x and determine if U is the midpoint of [pic]. (Important skill!)
Segment Bisectors
Plot two points Q and R on the number line and then draw [pic] through M, the midpoint of [pic].
Segment bisector: any segment, line, or plane that intersects a segment at its midpoint.
IMPLICATION: A segment bisector creates two congruent segments.
REMEMBER: Objects and figures are congruent. e.g. [pic]
Numbers and measures are equal. e.g. AB = BC
Practice:
In the figure, [pic] bisects [pic] at X
and [pic] bisects [pic] at Y. Given the
following conditions, find the value
of x and the measure of the indicated
segment.
a. AX = 2x + 11, XB = 4x – 5; [pic]
b. AB = x + 3, AX = 3x – 1; [pic]
c. YB = 23 – 2x, XY = 2x + 3; [pic]
d. AX = 27 – x, XB = 13 – 3x; [pic]
e. AB = 5x – 4, XY = x + 1: [pic]
( HW: A7a pp 26-27 #31-40, 43-44
A7b fms-Geometry Worksheet 1.3
( HW: A6-7 pp 25-27 #13-39 odd, 43-44
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