Question

Seth is using the figure shown below to prove the Pythagorean Theorem using triangle similarity:

In the given triangle DEF, angle D is 90° and segment DG is perpendicular to segment EF.
Part A: Identify a pair of similar triangles. (2 points)

Part B: Explain how you know the triangles from Part A are similar. (4 points)

Part C: If EG = 2 and EF = 8, find the length of segment ED. Show your work

Answer 1

In triangle DEF a perpendicular line is drawn from D to side EF (an altitude).

There is a theorem that states: If an altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and each other.

This means that triangle DEG is similar to DEF. The reason is AA (two angles congruent). Both triangles have a right angle (angle EDF in triangle DEF and angle EGD in triangle DEG) and both triangles contain angle E.

Another theorem states that in a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. Each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

This means the ED, a leg of the right triangle, is the geometric mean of EF, the length of the hypotenuse (8), and the EG, the segment adjacent to the leg (2)

ED being the geometric mean, implies that ED**2 = EF * EG --> ED**2 = 8 * 2 --> ED**2 = 16 --> ED = 4.

Answer 2

thank you sm!

Answer 3

you welcome