Question

The following sides of triangle upper A upper B upper C are labeled: upper A upper C is lower b, upper C upper B is lower a, and upper A upper B is lower c. Upper A upper O is labeled lower y and upper O upper B is labeled lower x. Right angle symbols are drawn at angle upper A upper C upper B and angle upper C upper O upper B.


Given is the right triangle ABC
with altitude CO
. Which one of the following statements is true and contributes to the proof of the Pythagorean Theorem?

(1 point)
Responses

△ABC∼△BOC

triangle upper A upper B upper C congruent to triangle upper A upper C upper O

△AOC∼△COB

triangle upper A upper B upper C congruent to triangle upper A upper C upper O

△AOC∼△BOC

triangle upper A upper B upper C congruent to triangle upper A upper C upper O

△ABC∼△AOC

Answer 1

△ABC ∼ △AOC.

Reason: ∠C and ∠O are right angles and ∠A is common, so the two triangles are similar. From the similarity AO/AC = AC/AB, i.e. y/b = b/c, so b^2 = c·y. Similarly △ABC ∼ △BOC gives a^2 = c·x. Adding gives a^2 + b^2 = c(x + y) = c^2, the Pythagorean relation.