To determine why the triangles are similar and to write the true proportions [tex]\(\frac{c}{a} = \frac{a}{f}\)[/tex] and [tex]\(\frac{c}{b} = \frac{b}{e}\)[/tex], we need to invoke the correct geometric theorem.
Given a right triangle, the right triangle altitude theorem states that if an altitude is drawn to the hypotenuse, it divides the right triangle into two smaller right triangles that are similar to each other and to the original triangle.
According to this theorem, the proportions involving the hypotenuse and the segments formed can be set up. Specifically:
1. Similarity: The two smaller triangles formed by the altitude are similar to the original right triangle and to each other.
2. Proportionality: From the similarity of triangles, we derive the proportions relating the sides of the triangles.
This allows us to write:
- [tex]\(\frac{c}{a} = \frac{a}{f}\)[/tex]
- [tex]\(\frac{c}{b} = \frac{b}{e}\)[/tex]
Here:
- [tex]\(c\)[/tex] is the length of the hypotenuse of the original right triangle.
- [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the lengths of the legs of the original right triangle.
- [tex]\(e\)[/tex] and [tex]\(f\)[/tex] are the segments of the hypotenuse divided by the altitude.
Therefore, the theorem that allows you to state that the triangles are similar in order to write the true proportions [tex]\(\frac{c}{a} = \frac{a}{f}\)[/tex] and [tex]\(\frac{c}{b} = \frac{b}{e}\)[/tex] is:
The right triangle altitude theorem.
