Question

Julia is using this figure to prove that triangle ABC is an isosceles triangle. First, she used the converse of the perpendicular bisector theorem and the definition of perpendicular lines to determine that CE is the perpendicular bisector of AB.

Answer 1

AC = BC because of the perpendicular bisector theorem

Explanation:

I got it right on the test

Answer 2

We have proven that AB = AC, which means that triangle ABC is isosceles.

The next step of a valid proof that triangle ABC is isosceles is to show that triangles ACE and BCE are congruent. This can be done using the following steps:

State the givens:

CE is the perpendicular bisector of AB.

Apply the Pythagorean Theorem to triangles ACE and BCE:

AC^2 = AE^2 + EC^2

BC^2 = BE^2 + EC^2

Subtract the two equations:

AC^2 - BC^2 = AE^2 - BE^2

Factor the left-hand side of the equation:

(AC + BC)(AC - BC) = AE^2 - BE^2

Rewrite the right-hand side of the equation using the difference of squares factorization:

(AC + BC)(AC - BC) = (AE + BE)(AE - BE)

Cancel out the common factor of (AE + BE):

AC + BC = AE - BE

This proves that triangles ACE and BCE are congruent, which means that AC = BC. Therefore, triangle ABC is isosceles.

Question