Given is the right triangle ABC

Question

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

(1 point)
Responses

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

△AOC∼△BOC

Answer 1

To prove the Pythagorean Theorem using similar triangles, we need to identify the relationships between the triangles formed in the right triangle with an altitude.

In triangle \( ABC \), if \( CO \) is the altitude from \( C \) to \( AB \), we can see that:

• Triangle \( AOC \) is similar to triangle \( ABC \) because they share angle \( A \) and both have a right angle.
• Triangle \( BOC \) is also similar to triangle \( ABC \) for the same reasons: they share angle \( B \) and both contain a right angle.
• Therefore, we have the following similarity relationships:
  • \( \triangle ABC \sim \triangle AOC \)
  • \( \triangle ABC \sim \triangle BOC \)

Given the options provided, the correct response that contributes to the proof of the Pythagorean Theorem is:

\( \triangle ABC \sim \triangle AOC \)

Thus, the statement that is true is:

△ABC∼△AOC.

This similarity leads to the relationships that help establish the Pythagorean theorem.