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. (4 points)

Answer 1

In triangle DEF, with a right angle at D and perpendicular segment DG to EF, triangles DGE and DEF are identified as similar by the Angle-Angle (AA) similarity criterion. Using the similarity, the length of segment ED is found as 8 units when EG is 2 and EF is 8.

**Part A: Identify a pair of similar triangles.**

Two pairs of similar triangles in the figure are:

1. Triangle DGE and triangle DEF

2. Triangle DGE and triangle DGF

**Part B: Explain how you know the triangles from Part A are similar.**

In both cases, the angles are the same: angle D is 90° in triangle DEF, and angle DGE is a right angle. Additionally, by the Angle-Angle (AA) similarity criterion, the other angles are equal. Thus, the two triangles are similar.

**Part C: Find the length of segment ED.**

Given that triangles DGE and DEF are similar, we can set up a proportion:

`\[ (ED)/(DG) = (EF)/(GE) \]`

Substitute the given values:

`\[ (ED)/(DG) = (8)/(2) \]`

Cross-multiply:

`\[ ED * 2 = DG * 8 \]`

Solve for ED:

`\[ ED = (DG * 8)/(2) \]`

If EG = 2 and EF = 8:

`\[ ED = (2 * 8)/(2) = 8 \]`

Therefore, the length of segment ED is 8 units.