Geometry Practice Test: Similar Triangles
- Similar Triangles
- 10 questions
- 14 minutes
These questions use similarity criteria and proportions between corresponding sides. Tip: write the similarity statement with matching vertex order before you set up any proportion.
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Question 1
In △ABC, m∠A = 48° and m∠B = 67°. In △DEF, m∠D = 48° and m∠F = 65°.
Which statement is correct?
Correct answer: B
m∠C = 180° − 48° − 67° = 65° and m∠E = 180° − 48° − 65° = 67°. So A matches D, B matches E and C matches F, and the triangles are similar by AA.
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Question 2
Two triangles are similar with a scale factor of 3 : 7 (small to large). The perimeter of the smaller triangle is 24 cm.
What is the perimeter of the larger triangle?
Correct answer: C
Perimeters scale by the same factor as sides: 24 · 7/3 = 56 cm. Adding 4 (the difference 7 − 3) to 24 gives the wrong answer 28.
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Question 3
Two similar triangles have areas of 18 square units and 50 square units.
What is the ratio of their corresponding sides (small to large)?
Correct answer: D
The ratio of areas is the square of the side ratio: 18 : 50 = 9 : 25, and 9/25 = 3/5. The ratio 9 : 25 is the area ratio itself.
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Question 4
In right triangle ABC, the right angle is at C. The altitude from C meets hypotenuse AB at D, with AD = 5 and DB = 15.
What is the length of leg AC?
Correct answer: A
Each leg is the geometric mean of the hypotenuse and the adjacent segment: AC² = AD · AB = 5 · 20 = 100, so AC = 10. Using 5 · 15 gives 53, which is the altitude CD, not the leg.
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Question 5
In △PQR, S is on PQ and T is on PR so that ST ∥ QR. PS = 8, SQ = x, PT = 10 and TR = 15.
What is the value of x?
Correct answer: B
By the Triangle Proportionality (side-splitter) Theorem, PS/SQ = PT/TR: 8/x = 10/15, so x = 12. The value 18.75 comes from flipping one of the ratios.
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Question 6
In △ABC, ray AD bisects ∠A and meets BC at D. AB = 9, AC = 12 and BC = 14.
What is BD?
Correct answer: C
The Angle Bisector Theorem gives BD/DC = AB/AC = 9/12 = 3/4. So BD = 14 · 3/7 = 6 and DC = 8. Choosing 7 assumes the bisector cuts the opposite side in half, which is only true when AB = AC.
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Question 7
A student places a small mirror flat on the ground 15 m from the base of a building. She stands 2 m from the mirror on the other side and sees the top of the building in it. Her eyes are 1.6 m above the ground.
How tall is the building?
Correct answer: B
The angles of incidence and reflection are equal, so the two right triangles are similar: h/15 = 1.6/2, which gives h = 12 m. The result 18.75 m comes from 15 · 2/1.6, which mixes up the ratio.
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Question 8
In △ABC, AB = 6, AC = 9 and m∠A = 50°. In △DEF, DE = 8, DF = 12 and m∠D = 50°.
Which statement is true?
Correct answer: D
The sides around the congruent angles are proportional (6/8 = 9/12 = 3/4) and the included angles are both 50°, so SAS similarity applies. Only one pair of angles is known, so AA cannot be used, and the third sides are not given, so SSS cannot be used.
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Question 9
A triangle has sides of 5, 7 and 9. A similar triangle has a longest side of 27.
What is the length of the shortest side of the larger triangle?
Correct answer: A
The scale factor is 27/9 = 3, so the shortest side is 5 · 3 = 15. Adding 18 (the difference 27 − 9) to 5 gives 23, but similar figures scale by multiplying, not adding.
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Question 10
In △ABC, D is on AB and E is on AC with DE ∥ BC. AD = 4, DB = 6 and BC = 15.
What is the length of DE?
Correct answer: A
△ADE ~ △ABC, so DE/BC = AD/AB = 4/10. Then DE = 15 · 0.4 = 6. Using AD/DB = 4/6 gives 10, but DB is not a side of either similar triangle.
Questions you leave blank are not counted as wrong.
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