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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

△ABC∼△AOC
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∼△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

Answer 1

To determine which statement is true and contributes to the proof of the Pythagorean Theorem based on the right triangle ABC with altitude CO, we can analyze the similarity of the triangles formed by the altitude to the hypotenuse.

In right triangle ABC, when you draw altitude CO to the hypotenuse AB, you create several relationships between the smaller triangles AOC, BOC, and the larger triangle ABC.

The correct statement that contributes to the proof of the Pythagorean Theorem is:

△AOC∼△COB

This indicates that triangle AOC is similar to triangle BOC. Since both of these triangles share angle C, and they are both right triangles (with angle O being the right angle), this similarity helps establish the relationships needed to derive the Pythagorean theorem, stating that \(a^2 + b^2 = c^2\), where a and b are the legs of the right triangle and c is the hypotenuse.

So the answer is:

△AOC∼△COB