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 } \\)?
\\( \bigcirc \\) the geometric mean (altitude) theorem
\\( \bigcirc \\) the geometric mean (leg) theorem
\\( \bigcirc \\) the right triangle altitude theorem
\\( \bigcirc \\) the sss theorem

Explanation:

The right - triangle altitude theorem (also known as the geometric mean theorem) states that if an altitude is drawn from the right angle of a right triangle to its hypotenuse, then the two triangles formed are similar to the original triangle and to each other. In the given problem, when we consider the right triangle \(ABC\) with right angle at \(C\) and altitude \(CD\), triangles \(ABC\), \(ACD\), and \(BCD\) are similar.

For the proportion \(\frac{c}{a}=\frac{a}{f}\), we can think of the relationship between the hypotenuse \(c\) of the large triangle \(ABC\) and the side \(a\) of triangle \(ABC\) and the side \(a\) of triangle \(BCD\) and its corresponding side \(f\). Similarly, for \(\frac{c}{b}=\frac{b}{e}\), we consider the relationship between the hypotenuse \(c\) of the large triangle \(ABC\) and the side \(b\) of triangle \(ABC\) and the side \(b\) of triangle \(ACD\) and its corresponding side \(e\).

The geometric mean (altitude) theorem is about the altitude and the segments of the hypotenuse (\(h^{2}=e\times f\)). The geometric mean (leg) theorem is not a standard name for the theorem used here. The SSS (Side - Side - Side) theorem is for proving triangle congruence (when the ratios of all three corresponding sides of two triangles are equal), not for the similarity of right - triangles formed by an altitude in a right - triangle.

Answer:

the right triangle altitude theorem