Question

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

Answer 1

In a right triangle \( ABC \) with the right angle at \( C \) and altitude \( CO \) drawn to hypotenuse \( AB \), the relationships between the triangles can be examined using similarity.

1. Triangle \( AOC \) and Triangle \( COB \): These triangles are similar because they share angle \( C \) and have a right angle. Therefore, \( \triangle AOC \sim \triangle COB \).

2. Triangle \( ABC \) and Triangle \( AOC \): Triangle \( ABC \) is also similar to triangle \( AOC \) because they share angle \( A \) and both have a right angle \( C \). Therefore, \( \triangle ABC \sim \triangle AOC \).

3. Triangle \( ABC \) and Triangle \( BOC \): Triangle \( ABC \) is similar to triangle \( BOC \) for the same reasons; they share angle \( B \) and both have a right angle \( C \). Hence, \( \triangle ABC \sim \triangle BOC \).

From the given statements, the correct choice that indicates a true relationship contributing to the proof of the Pythagorean Theorem is:

\(\triangle AOC \sim \triangle COB\)

This indicates the similarity in triangle \( AOC \) and triangle \( COB \).