Essentials of geometry Midpoints and Bisectors Ch 1.4 C. N. Colón Geometry St Barnabas HS Bronx, NY

Midpoint The point that bisects a segment. 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 = 2AM and AB = 2MB ~

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

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

Midpoint 12x+3 10x+5 A M B 12x+3=10x+5 2x=2 x=1

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

Add & Subtract line segments REMEMBER: A B x1 x2 The length of AB can be found by x2-x1 The symbol for the length of AB is AB.

Add & Subtract line segments P Q R S -4 -3 -2 -1 0 1 2 3 4 PQ = |(-1) – (-3)| = |-1 + 3| = 2 PR = |0 – (-3)| = |0 + 3| = 3 QR = |0 – (-1)| = |0 + 1| = 1 RS = | 3 – 0| = |3| = 3 QS = |3 – (-1)| = |3 + 1| = 4 PS = |3 – (-3)| = |3 + 3| = 6

Add & Subtract line segments P Q R S -4 -3 -2 -1 0 1 2 3 4 PR = RS and PR + RS = PS R is the midpoint of PS PQ + QS = 2 + 4 = 6 and PS = 6  PQ + QS = PS

Homework Read pp. 11-13, p. 13 # 1-12 (odd)