Question 1
A right triangle has legs of lengths 7 and 24. What is the length of the hypotenuse?
Enter your own answer.
Answer: 25
By the Pythagorean theorem, c² = 7² + 24² = 49 + 576 = 625, so c = 25.
SAT · Math · Geometry and Trigonometry
Right triangles and trigonometry
Right triangles and trigonometry is part of Geometry and Trigonometry in the SAT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.
Question types you will see here include missing side with the Pythagorean theorem, sine, cosine, tangent from side lengths, complementary angle identity and special right triangles.
Common traps: treating a leg as the hypotenuse; adding instead of subtracting squares; not taking the square root.
Sample questions
Question 2
An isosceles right triangle has a hypotenuse of length 12. What is the length of each leg?
- A.
6√2
- B.
6√3
- C.
12
- D.
12√2
Answer: A. 6√2
In a 45°-45°-90° triangle the hypotenuse is leg × √2, so each leg is 12/√2 = 6√2 (A). B uses the 30°-60°-90° long-leg ratio; C makes the leg as long as the hypotenuse; D multiplies by √2 instead of dividing.
- A.
Question 3
A sushi counter presses cooked rice into a rectangular tray that measures 24 cm by 10 cm. What is the length, in centimetres, of the tray's diagonal?
- A.
14
- B.
26
- C.
34
- D.
676
Answer: B. 26
The diagonal is the hypotenuse of a right triangle with legs 24 and 10, a multiple (×2) of the 5-12-13 triple, so it is 26 cm. A subtracts the sides. C adds the sides. D is the sum of the squares, 576 + 100, without taking the square root.
- A.
Question 4
A right triangle has a hypotenuse of length 58 and one leg of length 40. What is the length of the other leg?
Enter your own answer.
Answer: 42
The numbers 40 and 58 are twice 20 and 29 from the 20-21-29 triple, so the other leg is 2 · 21 = 42. Check: 40² + 42² = 1600 + 1764 = 3364 = 58². Subtracting 58 − 40 = 18 ignores the Pythagorean relationship, which applies to squares.
Question 5
For an acute angle θ in a right triangle, sin θ = 2/3. What is the value of tan θ?
- A.
√5/3
- B.
2√5/5
- C.
√5/2
- D.
3/2
Answer: B. 2√5/5
Take opposite 2 and hypotenuse 3; the adjacent side is √(9 − 4) = √5, so tan θ = 2/√5 = 2√5/5 (B). A is cos θ; C is adjacent over opposite, the reciprocal of tan θ; D is the hypotenuse over the opposite side.
- A.
Find your real gaps
A few questions show the style. The free scan shows which SAT skills cost you the most points, then builds a plan around them.