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?

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?
AOC ~ COB
ABC ~ AOC
ABC ~ BOC
AOC ~ BOC

GPT-4o mini · Answer

To analyze the triangle ABC with altitude CO, we can identify the relationships among the triangles formed.

In triangle ABC, we have:

AOC, which consists of the right triangle formed by the altitude from C to side AB.
BOC, which is the right triangle that also includes the altitude from C to AB.
Additionally, triangle ABC itself is a right triangle.

Given that triangles AOC and BOC both have a right angle (at C) and share angle C, we can see that triangles AOC and BOC are similar by the AA (Angle-Angle) similarity criterion.

Therefore, the true statement that contributes to the proof of the Pythagorean theorem is: AOC ~ BOC.

This statement represents that the triangles AOC and BOC are similar, leading to relationships among their sides, which ultimately contribute to proving the Pythagorean theorem.