What theorem did we prove using similar right triangles.

Pythagoras's theorem

Step-by-step explanation:

In the given drawing, there are three right triangles (two smaller triangles formed inside one right triangle), which are;

1) The smallest right triangle with side lengths 'b', 'f', and 'd'

2) The intermediate right triangle with side lengths 'f', 'a' and 'c - d'

3) The right triangle ΔABC containing the other two with side lengths, 'a', 'b', 'c'

By trigonometric ratio we have;

1/cos(B) = a/(c - d) = c/a

By cross multiplication, we get

a² = c² - c·d

Similarly, we have;

The vertex angle for the small right triangle at C = ∠B

1/sin(B) = b/d = c/b

∴ b² = c·d

Therefore, adding the values of 'a²', and 'b²' together, we get;

a² + b² = c² - c·d + c·d = c²

Therefore;

a² + b² = c² which is Pythagoras's theorem.

answer: (x+2) (x+3) are the binomials for your question : )