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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

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

△AOC∼△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

△ABC∼△BOC

Answer 1

To determine which statement contributes to the proof of the Pythagorean Theorem based on the description of triangle \( ABC \) and the altitude \( CO \), we need to analyze the relationships between the triangles.

In a right triangle, when an altitude is drawn from the right angle to the hypotenuse, it creates two smaller triangles. In this case, triangles \( AOC \) and \( BOC \) are created.

The key relationship to notice here is that triangles \( AOC \) and \( BOC \) are similar to triangles \( ABC \) due to the properties of right triangles and corresponding angles.

Given these relationships, the statement that best captures a valid similarity condition contributing to the proof of the Pythagorean Theorem is:

△AOC∼△BOC

This similarity implies that the ratios of the lengths of corresponding sides are equal, which can be used to establish the relationship between the areas of these triangles and the segments on the hypotenuse.

Therefore, the correct answer is:

△AOC∼△BOC