Midpoints and Bisectors Ch 1 4 C N

Midpoints and Bisectors Ch 1 -4 C. N. Colón Geometry St Barnabas HS Bronx, NY

The point that bisects a segment. Bisects ? means split into 2 equal or congruent segments A M B If M is the midpoint of AB, then AM ~= MB AM = ½ AB and MB = ½ AB AB = 2 AM and AB = 2 MB

A -3 M -2 -1 0 1 B 2 3 4 5 For any line segment AB, if the coordinate of A ihas a value of a, and the coordinate of B is has a value of b, then the coordinate of the midpoint of AB is a + b 2 -3 + 5 = 2 2 2 = 1, the midpoint M

ALWAYS REMEMBER! If objects are congruent, then you can set their measures equal to each other!

12 x+3 A 10 x+5 M 12 x+3 = 10 x+5 B 2 x = 2 x=1 PROOF: AM = 12 x + 3 = 12(1) + 3 = 12 + 3 = 15 MB = 10 x + 5 = 10(1) + 5 = 10 + 5 = 15

A bisector of a line segment is any segment, ray, line, or plane that intersects a segment at its midpoint. l A M B

REMEMBER: A B x 1 x 2 The length of AB can be found by x 2 -x 1 The symbol for the value or length of AB is AB.

R P S A line segment, RS is the sum of two line segments, RP and PS, if P is between R and S We can refer to the length of the segments and state the following equalities as true for their length. RS = RP + PS RP = RS – PS PS = RS – RP

P -4 -3 Q -2 -1 R 0 S 1 2 3 Find the length PQ of = each segment |(-1) – (-3)| = |-1 + 3| = 2 PR = |0 – (-3)| = |0 + 3| = 3 QR = RS = QS = PS = |0 – (-1)| = |0 + 1| = 1 | 3 – 0| = |3| = 3 |3 – (-1)| = |3 + 1| = 4 |3 – (-3)| = |3 + 3| = 6 4

P -4 -3 Q -2 -1 R 0 S 1 2 3 4 What point is the If PR = RS andmidpoint then R is the midpoint of PS Show that PR + RS = PS PQ + QS = 2 + 6 = PS PQ 4 +=QS PQ + QS = PS and PS = 6

p. 13 # 3 -12 (e)