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. The figure shows triangle DEF with right angle at D and segment DG. Point G is on side 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

Part A - Triangles DGE and DEF

Part B - Triangles DGE and DEF are similar because they have two congruent angles.

Part C - The length of segment ED is 2.

Part A: Identify a pair of similar triangles.

Answer: Triangles DGE and DEF

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

Answer: Triangles DGE and DEF are similar because they have two congruent angles.

Angle DGE is congruent to angle DEF because they are both right angles (all right angles are congruent).

Angle GDE is congruent to angle FED because they are both vertical angles (vertical angles are always congruent).

Therefore, by the Angle-Angle (AA) Similarity Criterion, we can conclude that triangles DGE and DEF are similar.

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

Answer: We can set up a proportion to find the length of segment ED.

EG:EF = DE:DG

Substituting in the known values, we get:

2:8 = DE:DG

We can cross-multiply to solve for DE:

2 * DG = 8 * DE

Since we know that DG is equal to 8 (because it is the length of segment EF), we can substitute that in to get:

2 * 8 = 8 * DE

16 = 8 * DE

DE = 2

Therefore, the length of segment ED is 2.