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Right triangles and trigonometry — SAT practice questions

Right triangles and trigonometry on the SAT Math test covers the Pythagorean theorem, special right triangles, and sine, cosine, and tangent. Students must classify a triangle from given side lengths, find a missing side or trigonometric value in a labeled figure, and find the perimeter of an isosceles right triangle cut from a square. Items often ask which statement describes the largest angle or ask for a numerical value. Choices may label an angle acute, right, or obtuse, or offer lengths that swap a leg with the hypotenuse. A common trap is using the wrong ratio.

  • Section: Math 44 questions on the paper
  • 68 practice questions in the app

Sample questions

Question 1

A triangle has side lengths 8, 15, and 17. Which of the following describes the largest angle of the triangle?

  1. AIt is acute.
  2. BIt is a right angle.
  3. CIt is obtuse.
  4. DIt cannot be determined.
Show answer

Answer: B — It is a right angle.

Since 82+152=1728^2+15^2=17^2, the angle opposite the side of length 17 is a right angle.

Question 2

A square with side length 31231\sqrt{2} units is cut along a diagonal, producing two isosceles right triangles. What is the perimeter, in units, of either triangle?

  1. A31+31231+31\sqrt{2}
  2. B62262\sqrt{2}
  3. C124
  4. D62+62262+62\sqrt{2}
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Answer: D — 62+62262+62\sqrt{2}

Each triangle has two legs of length 31231\sqrt{2}, and its hypotenuse, the square's diagonal, has length (312)(2)=62(31\sqrt{2})(\sqrt{2})=62. The perimeter is 622+6262\sqrt{2}+62.

Question 3

In right triangle ABCABC, angle BB is a right angle and sin⁡A=12\sin A=\dfrac{1}{2}. What is the value of cos⁡A\cos A?

  1. A12\dfrac{1}{2}
  2. B22\dfrac{\sqrt{2}}{2}
  3. C32\dfrac{\sqrt{3}}{2}
  4. D33\dfrac{\sqrt{3}}{3}
Show answer

Answer: C — 32\dfrac{\sqrt{3}}{2}

With opposite 1 and hypotenuse 2, the adjacent leg is 3\sqrt{3}, so cos⁡A=32\cos A=\dfrac{\sqrt3}{2}.

Practice 68 Right triangles and trigonometry questions in the app

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