Write a coordinate proof.

Question

Triangle ABC is a right isosceles triangle with hypotenuse AB.M is the midpoint of AB. Write a coordinate proof to prove that CM is perpendicular to AB. 

Answer 1

Δ ABC is right isosceles with hypotenuse AB

therefore AC=BC by definition of isosceles Δ

M is the midpoint of AB

therefore AM=MB by definition of midpoint

CM=CM reflexive property

ΔACM≅ΔBCM by SSS

<A=45º and <B=45º because of right isosceles triangle

<ACM=<BCM because they are corresponding angles of congruent triangles

<ACB=90º

<ACM+<BCM=<ACB because they are adjacent angles and The Angle Addition Postulate

<ACM+<ACM=<ACB substitution

2<ACM=90º

<ACM=45º division property of equality

<A=45º, <ACM=45º, so therefore <AMC=90º because the sum of the angles in a triangle is 180º

if <AMC=90º, then CM is perpendicular to AB because perpendicular lines form 90 degree angles