Test yourself on Geometry: Triangles with 10 original SAT practice questions. Pick an answer to see instant feedback and a full explanation.
Free original practice questions for study purposes. Open Exam Prep is an independent study resource and is not affiliated with, endorsed by, or sponsored by the makers of SAT.
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Answer the questions below — you get instant feedback and a full explanation for each.
1. In a right triangle, one leg measures 6 and the hypotenuse measures 10. What is the length of the other leg?
Explanation. By the Pythagorean theorem, leg² = 10² − 6² = 100 − 36 = 64, so the leg = 8. This is a 6-8-10 triangle (a scaled 3-4-5).
2. Two angles of a triangle measure 35° and 75°. What is the measure of the third angle?
Explanation. Angles in a triangle sum to 180°. So the third angle = 180 − 35 − 75 = 70°.
3. A triangle has sides of length 5, 12, and x. Which value of x is NOT possible?
Explanation. By the triangle inequality, x must satisfy 12 − 5 < x < 12 + 5, i.e., 7 < x < 17. Since 18 is not less than 17, it is impossible. The others all fall within the range.
4. In an equilateral triangle with side length 6, what is the area?
Explanation. Area of an equilateral triangle = (√3/4)s² = (√3/4)(36) = 9√3.
5. Triangle ABC is similar to triangle DEF. If AB = 4, DE = 6, and the area of ABC is 8, what is the area of DEF?
Explanation. For similar figures, the ratio of areas equals the square of the ratio of corresponding sides. The side ratio is 6/4 = 3/2, so area ratio = (3/2)² = 9/4. Area of DEF = 8 × 9/4 = 18.
6. A 30-60-90 triangle has its shortest side equal to 5. What is the length of the hypotenuse?
Explanation. In a 30-60-90 triangle, the sides are in ratio 1 : √3 : 2, where the shortest side (opposite 30°) is the '1'. The hypotenuse = 2 × 5 = 10.
7. In a 45-45-90 triangle, each leg measures 7. What is the length of the hypotenuse?
Explanation. In a 45-45-90 triangle, the sides are in ratio 1 : 1 : √2. The hypotenuse = leg × √2 = 7√2.
8. In triangle ABC, angle A = angle B. If side a (opposite A) = 9, what is side b (opposite B)?
Explanation. In a triangle, equal angles are opposite equal sides (isosceles triangle theorem). Since A = B, the sides opposite them are equal, so b = 9.
9. A triangle has a base of 10 and a corresponding height of 6. A second triangle has the same area but a base of 12. What is the height of the second triangle?
Explanation. Area of first = ½(10)(6) = 30. For the second, 30 = ½(12)(h), so 6h = 30, h = 5.
10. In right triangle ABC, the right angle is at C. If sin A = 3/5, what is cos A?
Explanation. sin A = opposite/hypotenuse = 3/5, suggesting a 3-4-5 triangle. The adjacent side is 4, so cos A = adjacent/hypotenuse = 4/5. (Or use sin²A + cos²A = 1: cos²A = 1 − 9/25 = 16/25.)
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FAQ
What special right triangles should I memorize for the SAT?
Memorize the 30-60-90 triangle (sides in ratio 1 : √3 : 2) and the 45-45-90 triangle (sides in ratio 1 : 1 : √2). Also recognize common Pythagorean triples like 3-4-5, 5-12-13, and 8-15-17, since they appear frequently and let you skip the Pythagorean theorem.
How does the triangle inequality work on the SAT?
The triangle inequality states that the sum of any two sides must be greater than the third side. For sides a and b, the third side x must satisfy |a − b| < x < a + b. SAT questions often ask which value of a missing side is or isn't possible.
What's the key fact about similar triangles for area problems?
If two triangles are similar with a side ratio of k, then their area ratio is k². So if sides scale by 3/2, areas scale by (3/2)² = 9/4. Don't confuse the side ratio with the area ratio—this is a common SAT trap.