Question

justifying steps in the proof of the pythagorean theorem
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:

Step1: Recall the geometric mean (altitude) theorem

The geometric mean (altitude) theorem states that in a right - triangle, the altitude drawn to the hypotenuse is the geometric mean between the segments of the hypotenuse. Also, each leg of the right - triangle is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to that leg.

Step2: Analyze the given proportions

The proportions \(\frac{c}{a}=\frac{a}{f}\) and \(\frac{c}{b}=\frac{b}{e}\) are based on the geometric mean (altitude) theorem. In a right - triangle \(ABC\) with right angle at \(C\) and altitude \(CD = h\) (where \(c=a + e\)), for the proportion \(\frac{c}{a}=\frac{a}{f}\), \(a\) is a leg of a smaller right - triangle and \(c\) is the hypotenuse of the larger right - triangle, and \(f\) is a segment of the hypotenuse. Similarly for \(\frac{c}{b}=\frac{b}{e}\).

Answer:

the geometric mean (altitude) theorem