Geometry Section1.3

Using Segments and Congruence

Distance and Midpoint Formula

What is midpoint?

The midpoint M of PQ is the point between P and Q such that PM = MQ.

How do you find the midpoint?

On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is (a + b)/2.

P M Q

1.) Find the midpoint of AC:

Examples:

0-5 6

(-5 + 6)/2 = ½

2.) If M is the midpoint of AZ, 2.) If M is the midpoint of AZ,

AM = 3x + 12 and MZ = 6x – 9; find AM = 3x + 12 and MZ = 6x – 9; find the measure of AM and MZ.the measure of AM and MZ.

3x + 12 = 6x – 93x + 12 = 6x – 9

21 = 3x21 = 3x

X = 7X = 7

AM = 33 MZ = 33AM = 33 MZ = 33

Q. How do you find the midpoint of 2 ordered pairs?

A. In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates (x1, y1) and (x2, y2) are ((x1 + x2)/2), (y1 + y2)/2)

Example:1.) Find the midpoint, M, of A(2, 8) and B(4, -4).

x = (2 + 4) ÷ 2 = 3

y = (8 + (-4)) ÷ 2 = 2

M = (3, 2)

2.) Find M if N(1, 3) is the midpoint of MP where the coordinates of P are (3, 6).

M = (-1, 0)

EXAMPLE 3 Use the Midpoint Formula

a. FIND MIDPOINT The endpoints of RS are R(1,–3) and S(4, 2). Find the coordinates of the midpoint M.

SOLUTION

EXAMPLE 2 Use algebra with segment lengths

STEP 1 Write and solve an equation. Use the fact that VM = MW.

VM = MW4x – 1 = 3x + 3

x – 1 = 3x = 4

Write equation.

Substitute.

Subtract 3x from each side.Add 1 to each side.

Point M is the midpoint of VW . Find the length of VM .ALGEBRA

EXAMPLE 2 Use algebra with segment lengths

STEP 2 Evaluate the expression for VM when x = 4.

VM = 4x – 1 = 4(4) – 1 = 15

So, the length of VM is 15.

Check: Because VM = MW, the length of MW should be 15. If you evaluate the expression for MW, you should find that MW = 15.

MW = 3x + 3 = 3(4) +3 = 15

Bisectors

What is a segment bisector?

- Any segment, line, or plane that intersects a segment at its midpoint.

A B C

M

N

If B is the midpoint of AC, then MN bisects AC.

In the skateboard design, VW bisects XY at point T, and XT = 39.9 cm. Find XY.

Skateboard

SOLUTION

EXAMPLE 1 Find segment lengths

Point T is the midpoint of XY . So, XT = TY = 39.9 cm.

XY = XT + TY= 39.9 + 39.9= 79.8 cm

Segment Addition PostulateSubstitute.

Add.

GUIDED PRACTICE for Examples 1 and 2

2.

In Exercises 1 and 2, identify the segment bisectorof PQ . Then find PQ.

line l ; 11 57

ANSWER

Distance Formula

The Distance Formula was developed from the Pythagorean Theorem

Where d = distance

x =x coordinate and y=y coordinate