Question

Given right triangle ABC with altitude CE. By the Right Triangle Altitude Theorem, which similarity statement is true?

△ABC ∼ △EBC

△ABC ∼ △AEC

△ABC ∼ △CBE

△ABC ∼ △BCE

(40 points)

Answer 1

Explanation:

By the Right Triangle Altitude Theorem, the two triangles formed by the altitude of a right triangle are similar to the original triangle and to each other.

In this case, triangle ABC is the original right triangle and CE is the altitude. Therefore, the similarity statement that is true is:

△ABC ∼ △AEC

This is because triangles ABC and AEC are both right triangles that share the angle at C, and the altitude CE creates two pairs of congruent angles (the right angles and the angle at A and E) and a pair of proportional sides (CE and AE are corresponding altitudes).

So, we can write the similarity statement as:

△ABC ~ △AEC