Question

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

a) Part A: Identify a pair of similar triangles.
i) Triangle DEF and Triangle DGE
ii) Triangle DGE and Triangle EFG
iii) Triangle DEF and Triangle EFG
iv) Triangle DGE and Triangle DEF

b) Part B: Explain how you know the triangles from Part A are similar.
i) They have equal angles.
ii) They have proportional sides.
iii) Both i and ii.
iv) None of the above.

c) Part C: If EG = 2 and EF = 8, find the length of segment ED. Show your work.
a) ED = 4
b) ED = 6
c) ED = 10
d) ED = 12

Answer 1

Triangle DEF and Triangle DGE are similar because they have equal angles and proportional sides. To find ED given EG=2 and EF=8, we use the Pythagorean theorem: DE² = 8² - 2², resulting in DE being approximately 7.75, which doesn't match with any of the provided options.

Step-by-step explanation:

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

Part A: Identify a pair of similar triangles.

The correct pair of similar triangles is Triangle DEF and Triangle DGE.

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

The triangles are similar because they have equal angles and they have proportional sides, hence option iii) Both i and ii is the correct answer.

Part C: Find the length of segment ED if EG = 2 and EF = 8.

To find the length of segment ED, we can use the following relationship derived from the Pythagorean theorem:

DE² = EF² - GE²

DE² = 8² - 2²

DE² = 64 - 4

DE² = 60

DE = √60

DE = 2√15 which is approximately 7.75, so none of the provided options are correct.

Answer 2

The pair of similar triangles is Triangle DEF and Triangle DGE. They are similar because they have equal angles and proportional sides. Using the property of similar triangles and the Pythagorean theorem, the length of segment ED is determined to be 8.

Step-by-step explanation:

The student's question pertains to the use of similar triangles and the Pythagorean theorem to find a missing side length in a right triangle.

Part A:

The pair of similar triangles can be identified as Triangle DEF and Triangle DGE.

Part B:

The triangles from Part A are similar because they have two angles in common (angle at D is shared, and each triangle has a right angle), thus by AA (angle-angle) similarity, the two triangles are similar. Consequently, they have both equal angles (i) and proportional sides (ii), so the correct answer is (iii) Both i and ii.

Part C:

To find the length of segment ED when EG = 2 and EF = 8, we use the property of similar triangles, which states that corresponding sides are in proportion. The ratio of EG to EF in Triangle DGE is the same as ED to EF in Triangle DEF. Thus, ED/EF = EG/EF; substituting the values we get ED/8 = 2/8, hence ED = 2. By the Pythagorean Theorem, in Triangle DEF, we have (ED)^2 + (EF)^2 = (DF)^2, substituting in the values we know (2)^2 + (8)^2 = (DF)^2, and solving for DF, we get DF = 8, which indicates Triangle DEF is an isosceles right triangle and ED, therefore, is also 8.