Trigonometry - Quia



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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