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Math: Geometry and Trigonometry, 3 original questions

SAT Right Triangle and Trigonometry Practice: Questions and Answers

Practice missing right-triangle sides, tangent ratios and special triangles in three original SAT-style questions, with exact answers and explained choices.

On this page

The short answer

For a right triangle, first locate the right angle and the hypotenuse opposite it. Use a² + b² = c² for side lengths, with c reserved for the hypotenuse. For an acute angle, identify its opposite and adjacent legs before choosing sine, cosine or tangent. Special right triangles can supply exact side ratios without measuring the diagram. Keep radicals or fractions exact when possible, and check that the hypotenuse is the longest side before accepting a result.

Method at a glance

  1. 1Establish the right-angle condition
  2. 2Choose a side or angle relationship
  3. 3Use a special triangle when given
  4. 4Check labels and exact values

Try the 3 questionsRead the full method

Step-by-step method

Use these steps in order, then apply them to the questions below.

  1. Step 1: Establish the right-angle condition

    Do not apply the Pythagorean theorem merely because a drawing looks right. Identify the given right angle and opposite hypotenuse.

  2. Step 2: Choose a side or angle relationship

    Use a² + b² = c² for lengths. For an acute angle, sine is opposite/hypotenuse, cosine is adjacent/hypotenuse and tangent is opposite/adjacent.

  3. Step 3: Use a special triangle when given

    A 30–60–90 triangle has side ratios 1:√3:2. A 45–45–90 triangle has side ratios 1:1:√2.

  4. Step 4: Check labels and exact values

    Opposite and adjacent depend on the chosen angle. Keep exact fractions and radicals unless the problem requests rounding.

Try 3 SAT-style questions

These are original SAT-style questions written by PeakSAT, not official College Board questions. Choose an answer and check it to see why each choice is right or wrong, or open the worked answer.

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Question 1 of 3

A right triangle has hypotenuse 13 units and one leg 5 units. What is the length of its other leg?

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Correct answer

Choice B: 12 units

Explanation

Let the other leg be b. Then 5² + b² = 13², so b² = 169 − 25 = 144. Since a side length is positive, b = 12. Checking gives 25 + 144 = 169.

Why each choice is right or wrong

  1. Choice A

    This subtracts side lengths instead of their squares.

  2. Choice B

    Correct answer

    Correct: √(13² − 5²) = 12.

  3. Choice C

    This adds the squares instead of subtracting the known leg's square from the hypotenuse's square.

  4. Choice D

    A leg cannot be longer than the hypotenuse 13.

Question 2 of 3

For an acute angle θ in a right triangle, sin θ = 5/13 and cos θ = 12/13. What is tan θ?

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Correct answer

Choice C: 5/12

Explanation

Tangent is opposite/adjacent. Equivalently, tan θ = sin θ / cos θ = (5/13)/(12/13) = 5/12. The shared hypotenuse factor cancels.

Why each choice is right or wrong

  1. Choice A

    This is the given sine ratio, not tangent.

  2. Choice B

    This reverses the opposite and adjacent legs, giving the reciprocal of tangent.

  3. Choice C

    Correct answer

    Correct: dividing sine by cosine cancels the shared hypotenuse and gives 5/12.

  4. Choice D

    This is the given cosine ratio, not tangent.

Question 3 of 3

In a 30–60–90 triangle, the shorter leg is 7√3 units. What is the longer leg?

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Correct answer

Choice C: 21 units

Explanation

The longer leg is √3 times the shorter leg. Therefore it is 7√3 × √3 = 7 × 3 = 21 units. The hypotenuse is 2 × 7√3 = 14√3 units, which is a different side.

Why each choice is right or wrong

  1. Choice A

    This divides by √3 rather than multiplying to obtain the longer leg.

  2. Choice B

    This drops the radical and uses an incorrect scale factor.

  3. Choice C

    Correct answer

    Correct: multiplying the shorter leg by √3 gives 21.

  4. Choice D

    This is the hypotenuse, which is twice the shorter leg.

Common mistakes

  • Using the hypotenuse as a leg in a² + b² = c². The side opposite the right angle belongs in c.

  • Using sine when the requested ratio is tangent. Tangent divides opposite by adjacent, not by hypotenuse.

  • Treating the longer leg of a 30–60–90 triangle as twice the shorter leg. It is √3 times the shorter leg; the hypotenuse is twice it.

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Common questions

Does the Pythagorean theorem work for every triangle?

No. It requires a right triangle. Use the stated angle information rather than a diagram's appearance.

How do I identify the opposite side?

Choose the angle first. Its opposite side is across from that angle; the adjacent leg touches it and is not the hypotenuse.

Should I convert an exact radical to a decimal?

Keep the exact form when possible. Round only when the question requests a decimal or specified precision.

Sources

  1. College Board: Geometry and Trigonometrysatsuite.collegeboard.org
  2. College Board: Student Question Bank Math skillssatsuite.collegeboard.org

Sources are linked for reference; they do not endorse PeakSAT. How we prepare our guides