Ch 2 Sect 1 Segment Bisectors The midpoint of a segment is the point that divides a segment into 2 congruent segments. Always look for markings to indicate midpoints. When working problems, make marks on quizzes and tests to remember where midpoints are located. Keep these relationships in mind when working with midpoints: Left + Right = Whole Left = Right Whole = small 2 (small) = whole 2 1

Ch 2 Sect 1 Segment Bisectors To bisect something means to divide it into 2 congruent parts. A segment bisector is a segment, line or ray that intersects a segment at its midpoint. (cuts it in half) 2

M is the midpoint of AB. Find AM and MB. Example 1 Find Segment Lengths M is the midpoint of AB. Find AM and MB. M is the midpoint of AB, so AM and MB are each half the length of AB . SOLUTION AM = MB = 2 1 · AB = · 26 13 ANSWER AM = 13 and MB = 13. 3

P is the midpoint of RS. Find PS and RS. Example 2 Find Segment Lengths P is the midpoint of RS. Find PS and RS. SOLUTION P is the midpoint of RS, so PS = RP. Therefore, PS = 7. You know that RS is twice the length of RP. RS = = = 4 2 · RP 2 · 7 ANSWER PS = 7 and RS = 14. 4

Checkpoint 1. Find DE and EF. ANSWER DE = 9; EF = 9 2. Find NP and MP. Find Segment Lengths Find DE and EF. 1. ANSWER DE = 9; EF = 9 Find NP and MP. 2. ANSWER NP = 11; MP = 22

Line l is a segment bisector of AB. Find the value of x. Example 3 Use Algebra with Segment Lengths Line l is a segment bisector of AB. Find the value of x. SOLUTION AM = MB Line l bisects AB at point M. Substitute 5x for AM and 35 for MB. 5x = 35 Divide each side by 5. 5 5x = 35 x = 7 Simplify. Check your solution by substituting 7 for x. 5x = 5(7) = 35 CHECK 6

Ch 2 Sect 1 ● A ● M ● B Segment Bisectors M is the midpoint of AB. Find y. 2y + 5 27 ● A ● M ● B 7

Ch 2 Sect 1 M = 2 x1 + x2 , y1 + y2 Segment Bisectors When we work with segments in the coordinate plane, we are usually given the coordinates of the endpoints of the segment. Our job is to find the coordinates of the exact midpoint. There is a formula we can use to do this. Midpoint Formula: Suppose our endpoints are A(x1, y1) and B(x2, y2). Then M will be the midpoint if we use M = 2 x1 + x2 , y1 + y2 8

Find the coordinates of the midpoint of AB. Example 4 Use the Midpoint Formula Find the coordinates of the midpoint of AB. A(1, 2), B(7, 4) a. SOLUTION First make a sketch. Then use the Midpoint Formula. a. Let (x1, y1) = (1, 2) and (x2, y2) = (7, 4). M = 2 x1 + x2 , y1 + y2 = 2 1 + 7 , 2 + 4 = (4, 3) 9

Let (x1, y1) = (–2, 3) and (x2, y2) = (5, –1). M = 2 x1 + x2 , y1 + y2 Example 4 Use the Midpoint Formula A(–2, 3), B(5, –1) b. b. Let (x1, y1) = (–2, 3) and (x2, y2) = (5, –1). M = 2 x1 + x2 , y1 + y2 = 2 –2 + 5 , 3 +(– 1) 2 3 , 1 = 10

Sketch PQ. Then find the coordinates of its midpoint. Checkpoint Use the Midpoint Formula Sketch PQ. Then find the coordinates of its midpoint. P(2, 5), Q(4, 3) 3. ANSWER (3, 4) P(0, –2), Q(4, 0) 4. ANSWER (2, –1) ANSWER – , 2 5 3 P(–1, 2), Q(–4, 1) 5.

Ch 2 Sect 1 Segment Bisectors Homework: page 56-57 #1-32 12