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
𝐴
𝐵
𝐶
with altitude CO
𝐶
𝑂
. Which one of the following statements is true and contributes to the proof of the Pythagorean Theorem?

(1 point)
Responses

△AOC∼△COB
△
𝐴
𝑂
𝐶
∼
△
𝐶
𝑂
𝐵

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

△ABC∼△AOC
△
𝐴
𝐵
𝐶
∼
△
𝐴
𝑂
𝐶

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

△ABC∼△BOC
△
𝐴
𝐵
𝐶
∼
△
𝐵
𝑂
𝐶

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

△AOC∼△BOC

Answer 1

Answer: △ABC ∼ △AOC.

Reason: Triangles ABC and AOC are similar (they share angle A and both are right triangles), so
AB/AC = AC/AO ⇒ c/b = b/y ⇒ b^2 = c·y.
Similarly, △ABC ∼ △BOC gives a^2 = c·x. Adding these,
a^2 + b^2 = c·x + c·y = c(x + y) = c·c = c^2,
which is the Pythagorean Theorem.