Question

Use the image to answer the question.

On right triangle upper A upper B upper C, the altitude of vertex upper C is drawn as a solid line that extends from upper C to the hypotenuse, upper A upper B. The intersection of the altitude and upper A upper B is labeled upper O.

Given is the right triangle ABC
with altitude CO
. Which statement, once proven, will imply that a2=cx
?

Statement #1: △ABC≅△ACO

Statement #2: △ABC≅△CBO

Statement #3: △ACO≅△CBO

Answer 1

Statement #1.

Reason: △ABC and △ACO are similar (they share ∠A and both have a right angle), so corresponding sides give
AC/AB = AO/AC.
Rewriting gives AC^2 = AB·AO. With a = AC, c = AB, and x = AO, this is a^2 = c x.

(Strictly speaking the triangles are similar, not congruent; the needed result follows from that similarity.)