Question

in a proof of the pythagorean theorem using similarity, what allows you to state that the triangles are similar in order to write the true proportions \\( \frac{c}{a}=\frac{a}{f} \\) and \\( \frac{c}{b}=\frac{b}{e} \\)?

• the geometric mean (altitude) theorem
• the geometric mean (leg) theorem
• the right triangle altitude theorem
• the sss theorem

Explanation:

The right - triangle altitude theorem (or geometric mean theorem) states that in a right triangle, the altitude drawn to the hypotenuse creates two similar right triangles that are also similar to the original right triangle. In the given problem, when we have a right triangle \(ABC\) with right angle at \(C\) and altitude \(CD\) to hypotenuse \(AB\), \(\triangle ABC\sim\triangle ACD\) and \(\triangle ABC\sim\triangle CBD\). From the similarity of \(\triangle ABC\) and \(\triangle ACD\), we get \(\frac{c}{b}=\frac{b}{e}\) (where \(c\) is the hypotenuse of \(\triangle ABC\), \(b\) is a leg of \(\triangle ABC\) and \(e\) is a segment of the hypotenuse \(AB\) ). From the similarity of \(\triangle ABC\) and \(\triangle CBD\), we get \(\frac{c}{a}=\frac{a}{f}\) (where \(a\) is a leg of \(\triangle ABC\) and \(f\) is the other segment of the hypotenuse \(AB\)). The geometric mean (altitude) theorem specifically refers to the relationship \(h^{2}=mn\) (where \(h\) is the altitude and \(m\) and \(n\) are the segments of the hypotenuse). The geometric mean (leg) theorem is related to the legs and the segments of the hypotenuse (\(x^{2}=m\times(m + n)\) for a leg \(x\)). The SSS (Side - Side - Side) theorem is for proving triangle congruence (not directly for the similarity in the context of the altitude of a right - triangle and the proportion setup here). So the right triangle altitude theorem is the most appropriate.

Answer:

the right triangle altitude theorem