Middle School Math Quiz: Triangle Similarity
4 questions · exam conditions
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Triangle SimilarityQuestion 1 of 4

Triangle ABC has sides of length 9, 12, and 15. Triangle XYZ has sides of length 6, 8, and k. If the triangles are similar, what is the value of k?

9
10
11
12
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Middle School Math Quiz

Middle School Math Quiz: Triangle Similarity

Practice Triangle Similarity in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Triangle Similarity, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Triangle ABC has sides of length 9, 12, and 15. Triangle XYZ has sides of length 6, 8, and k. If the triangles are similar, what is the value of k?

  1. 9
  2. 10 (correct answer)
  3. 11
  4. 12
Explanation: For the triangles to be similar, the ratios of corresponding sides must be equal. We need to determine which sides correspond. The ratio 6/9 = 2/3 and 8/12 = 2/3, so these pairs correspond. Therefore, k corresponds to the side of length 15, and k = 15 × (2/3) = 10. Choice A assumes k corresponds to the side of length 9 with a different ratio. Choice C uses an incorrect ratio calculation. Choice D assumes k corresponds to the side of length 12 with ratio 1:1.

Question 2

Triangle PQR has sides PQ = 20, QR = 24, and PR = 16. Triangle STU is similar to triangle PQR with a scale factor of 3/4 (smaller to larger). If the perimeter of triangle STU is 45, which side of triangle PQR corresponds to the longest side of triangle STU?

  1. PQ
  2. QR (correct answer)
  3. PR
  4. Cannot be determined from the given information
Explanation: First, verify the scale factor: Triangle PQR has perimeter 20 + 24 + 16 = 60. If triangle STU has perimeter 45, then the ratio is 45/60 = 3/4, confirming STU is the smaller triangle. The longest side of triangle PQR is QR = 24. Since the triangles are similar, the longest side of triangle STU corresponds to the longest side of triangle PQR, which is QR. Choice A corresponds to the medium side. Choice C corresponds to the shortest side. Choice D is incorrect because similarity preserves the relative ordering of side lengths.

Question 3

Two triangles have angles measuring 45°, 65°, and 70°. The first triangle has sides of length 8, 10, and 12. If the shortest side of the second triangle is 6, what is the length of the longest side of the second triangle?

  1. 8.0
  2. 9.0 (correct answer)
  3. 10.5
  4. 12.0
Explanation: Since both triangles have the same three angles, they are similar by AA similarity. The shortest side of the first triangle is 8, and the shortest side of the second triangle is 6, so the ratio is 6/8 = 3/4. The longest side of the first triangle is 12, so the longest side of the second triangle is 12 × (3/4) = 9. Choice A incorrectly uses the middle side. Choice C uses an incorrect ratio of 7/8. Choice D assumes the triangles are congruent.

Question 4

Two similar triangles have a ratio of corresponding sides of 5:3. If the area of the larger triangle is 100 square units, what is the area of the smaller triangle?

  1. 36 square units (correct answer)
  2. 45 square units
  3. 60 square units
  4. 75 square units
Explanation: When triangles are similar, the ratio of their areas equals the square of the ratio of their corresponding sides. The ratio of sides is 5:3, so the ratio of areas is 5²:3² = 25:9. If the larger triangle has area 100, then 100/x = 25/9, where x is the area of the smaller triangle. Solving: x = 100 × 9/25 = 36. Choice C uses the linear ratio instead of the squared ratio. Choice B and D represent other common calculation errors.