Similar Triangles Proportions Test Answer Key with Scoring – Geometry

Question 1 Similar triangle ratios (2 points)

The scale factor is XY ÷ PQ = 15 ÷ 9 = 5/3. Then YZ = (5/3)(12) = 20.

k = 5/3; YZ = 20

Scoring: 1 point per part.

Question 2 Choose the criterion (3 points)

1. Triangle 1 has angles 55°, 80°, 45°. Triangle 2 has angles 55°, 45°, 80°. Similar by AA.
2. Ratios: 6/4 = 1.5, 9/6 = 1.5, 13.5/9 = 1.5. Similar by SSS ∼.
3. 5/10 = 0.5 but 8/12 ≈ 0.667. The included sides are not proportional, so the triangles are not similar.

(a) similar (AA), (b) similar (SSS ∼), (c) not similar

Scoring: 1 point per part.

Question 3 Side-splitter lengths (3 points)

1. 6/10 = 9/EC, so EC = 90 ÷ 6 = 15.
2. AC = 9 + 15 = 24.
3. AB = 16, and △ADE ∼ △ABC gives DE/32 = 6/16, so DE = 12.

EC = 15; AC = 24; DE = 12

Scoring: 1 point per part.

Question 4 Converse of the side-splitter theorem (2 points)

4.5/6 = 0.75 and 6/8 = 0.75. The two sides are divided proportionally, so by the converse of the Side-Splitter Theorem, DE ∥ BC.

Yes, parallel

Scoring: 1 point for the two equal ratios, 1 point for the conclusion with the theorem.

Question 5 Altitude to the hypotenuse (3 points)

1. Area = ½ · 9 · 12 = 54 = ½ · 15 · h, so h = 108 ÷ 15 = 7.2.
2. 9² ÷ 15 = 5.4 and 12² ÷ 15 = 9.6; they add to 15.
3. 5.4 · 9.6 = 51.84 and 7.2² = 51.84.

h = 7.2; pieces 5.4 and 9.6

Scoring: 1 point per part.

Question 6 Prove similar triangles (3 points)

Scoring: 1 point for each correct pair of angles with the reason, 1 point for the AA conclusion.

Question 7 Mirror measurement (2 points)

The two right triangles are similar, so h/40 = 1.5/3. Then h = 40 · 1.5 ÷ 3 = 20.

The tower is 20 meters tall

Scoring: 1 point for the proportion, 1 point for the height.

Question 8 Parallel lines and transversals (2 points)

Parallel lines divide transversals proportionally: 80/120 = 70/x. So x = 70 · 120 ÷ 80 = 105.

105 meters

Scoring: 1 point for the proportion, 1 point for the answer.