Ask a New Question
Menu

Use the image to answer the 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

△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

1 answer

To analyze the given statements and determine which one is true and contributes to the proof of the Pythagorean Theorem, we should look at the relationships between the triangles formed by the altitude \(CO\) in triangle \(ABC\).

  1. △AOC∼△COB: This statement states that triangles \(AOC\) and \(COB\) are similar. This is indeed true. By the properties of similar triangles, corresponding angles are equal, and their sides are proportional.

  2. triangle ABC congruent to triangle ACO: This statement claims congruence between triangles \(ABC\) and \(ACO\), which isn't necessarily true since they share only one side.

  3. △ABC∼△AOC: This statement suggests that triangle \(ABC\) is similar to triangle \(AOC\). This is not correct because they do not have the same angles.

  4. triangle ABC congruent to triangle ACO (again): As mentioned earlier, this isn't correct.

  5. △AOC∼△BOC: This statement is also true. Triangles \(AOC\) and \(BOC\) are similar.

  6. triangle ABC congruent to triangle ACO (repeated): Again, this isn't true.

  7. △ABC∼△BOC: This statement suggests similarity between triangle \(ABC\) and triangle \(BOC\), which is also not true.

Out of these statements, the true and meaningful relationships that contribute to proving the Pythagorean theorem are:

  • △AOC∼△COB
  • △AOC∼△BOC

But since △AOC∼△COB is the original valid statement that aligns best with a common proof structure for the Pythagorean theorem (using similar triangles), this is likely the answer that best fits the request.

So the choice we would consider to be valid and contribute to the proof of the Pythagorean theorem is:

△AOC∼△COB.