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

 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 = 2AM and AB = 2MB ~

A M B a + b 2 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 = = 1, the midpoint M 2222

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

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

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

REMEMBER: AB x1x1x1x1 x2x2x2x2 The length of AB can be found by x 2 -x 1 The symbol for the value or length of AB is AB.

RPS 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

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 PQRS Find the length of each segment

If PR = RS and PR + RS = PS then R is the midpoint of PS PQ + QS = = 6 and PS = 6  PQ + QS = PS What point is the midpoint of PS Show that PQ + QS = PS PQRS

p. 13 # 3-12 (e)