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: 0-56 (-5 + 6)/2 = ½ 2.) If M is the midpoint of AZ, AM = 3x + 12 and MZ = 6x – 9; find the measure of AM and MZ. 3x + 12 = 6x – 9 21 = 3x X = 7 AM = 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 = MW 4x – 1 = 3x + 3 x – 1 = 3 x = 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. ABC 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 = = 79.8 cm Segment Addition Postulate Substitute. Add.

GUIDED PRACTICE for Examples 1 and 2 2. In Exercises 1 and 2, identify the segment bisector of PQ. Then find PQ. line l ; ANSWER

Distance Formula The Distance Formula was developed from the Pythagorean Theorem Where d = distance x =x coordinate and y=y coordinate