To determine which theorem allows us to state that the triangles are similar in the context of the Pythagorean theorem using similarity, we must consider the specific properties and relationships established by each theorem.
The correct theorem is the right triangle altitude theorem. This theorem states the following:
- In a right triangle, the altitude drawn to the hypotenuse creates two smaller right triangles that are each similar to the original right triangle and to each other.
By applying this theorem, we understand why the triangles are similar, allowing us to write the true proportions:
[tex]\[
\frac{c}{a} = \frac{a}{f}
\][/tex]
and
[tex]\[
\frac{c}{b} = \frac{b}{e}.
\][/tex]
Here’s a detailed explanation of why:
1. Right triangle altitude theorem - When an altitude is drawn from the right-angle vertex of a right triangle to the hypotenuse, it creates two smaller right triangles.
2. Similarity of triangles - Each of these smaller triangles is similar to the original triangle by the AA (Angle-Angle) similarity criterion because:
- Each smaller triangle shares an angle with the original triangle.
- All three triangles have a right angle.
3. Proportions based on similarity - Because the triangles are similar, corresponding sides are proportional. This allows us to write the proportions involving their sides:
- For [tex]$\frac{c}{a} = \frac{a}{f}$[/tex], side [tex]$c$[/tex] (hypotenuse of the original large triangle) is proportional to side [tex]$a$[/tex] (one leg of the small triangle) just as side [tex]$a$[/tex] (one leg of the large triangle) is proportional to side [tex]$f$[/tex] (corresponding leg of the small triangle).
- For [tex]$\frac{c}{b} = \frac{b}{e}$[/tex], side [tex]$c$[/tex] (hypotenuse of the original large triangle) is proportional to side [tex]$b$[/tex] (one leg of the other small triangle) just as side [tex]$b$[/tex] (one leg of the large triangle) is proportional to side [tex]$e$[/tex] (corresponding leg of the other small triangle).
The conclusion is that the right triangle altitude theorem grants the basis to assert the similarity of the triangles, allowing us to correctly write these proportions.
So, the correct answer is: the right triangle altitude theorem.
