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:

To determine which theorem allows stating the triangles are similar for the Pythagorean theorem proof using similarity, we analyze each option:

• The geometric mean (altitude) theorem relates the altitude to the segments of the hypotenuse, not directly to triangle similarity for these proportions.
• The geometric mean (leg) theorem involves the leg as a geometric mean of the hypotenuse and its adjacent segment, but it is derived from triangle similarity, not the basis for proving similarity.
• The right triangle altitude theorem (also called the geometric mean theorem or altitude-on-hypotenuse theorem) states that when an altitude is drawn to the hypotenuse of a right triangle, the two smaller right triangles formed are similar to the original triangle and to each other. This similarity (by AA criterion, as all right triangles have a right angle and share an acute angle) allows writing proportions like \(\frac{c}{a}=\frac{a}{f}\) and \(\frac{c}{b}=\frac{b}{e}\).
• The SSS theorem is for proving triangle congruence (or similarity via SSS similarity), but it does not apply here, as we use angle - angle (AA) similarity from the right triangle altitude theorem.

Answer:

the right triangle altitude theorem