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 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

△ABC∼△AOC
triangle upper A upper B upper C congruent to triangle upper A upper C upper O

△AOC∼△BOC

1 answer

To determine which statement is true and contributes to the proof of the Pythagorean Theorem, we need to analyze the relationship between the triangles formed by the altitude \( CO \) in triangle \( ABC \).

Given triangle \( ABC \) is a right triangle, when altitude \( CO \) is drawn to the hypotenuse \( AB \), it creates two smaller right triangles: \( AOC \) and \( BOC \).

The correct relationships among these triangles are:

Triangles \( ABC \) and \( AOC \) are similar because they share angle \( A \) and both have a right angle.
Triangles \( ABC \) and \( BOC \) are similar because they share angle \( B \) and both have a right angle.
Triangles \( AOC \) and \( BOC \) are also similar because they both share the angle \( C \) (the right angle), and they have a common segment \( CO \) and a side on the hypotenuse.

From these observations, let's break down the provided statements:

\( \triangle ABC \sim \triangle BOC \) (not provided, but true).
\( \triangle ABC \sim \triangle AOC \) (true).
\( \triangle AOC \sim \triangle BOC \) (true).

Among the provided potentially misleading options, the statement \( \triangle AOC \sim \triangle BOC \) is indeed true and is generally used in proofs related to the Pythagorean theorem.

Thus, the correct answer is:

△AOC∼△BOC