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ACT Right triangles & trigonometry practice questions

Right triangles & trigonometry sits in the Geometry reporting category on the ACT Math section. Prep36 has 16 questions tagged to this skill, every one with a worked explanation. Three of them are below in full, with the reasoning, so you can see what the skill actually looks like before you practise it.

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What Right triangles & trigonometry covers on the ACT

Within this skill the Prep36 bank covers Pythagorean theorem, cosine in a right triangle, sine ratio setup (sohcahtoa), law of sines, unit circle & quadrant signs and tangent and angle of elevation.

Math is 45 questions in 50 minutes, which is about 66 seconds each, and it is one of the three sections that average into your composite. The difficulty split across the 16 questions tagged here is 2 easier, 8 medium and 6 harder.

Questions are original, written to the Enhanced ACT format. Prep36 is not affiliated with ACT, Inc.

3 right triangles & trigonometry questions, with the reasoning

Question 1 easy

In a right triangle, the two legs measure 9 centimeters and 12 centimeters. What is the length, in centimeters, of the hypotenuse?

  1. A. 15 correct
  2. B. 21
  3. C. 108
  4. D. 225
Why: The hypotenuse comes straight from the Pythagorean theorem: c² = 9² + 12² = 81 + 144 = 225, so c = √225 = 15. This is just the 3-4-5 triple scaled by 3. The other three all skip or fumble a step: 225 is c² before you take the square root, 108 is the product 9 × 12, and 21 is the two legs added, which would only be the distance if the triangle flattened into a straight line. Tip: whenever you square things to solve, circle the variable you actually want so you remember to undo the square at the end.
Question 2 medium

A straight 20-foot ladder leans against a vertical wall and makes a 65° angle with the level ground. Which of the following is closest to the distance, in feet, from the base of the wall to the foot of the ladder?

  1. A. 8.5 correct
  2. B. 18.1
  3. C. 42.9
  4. D. 47.3
Why: The ladder is the hypotenuse and the ground distance sits adjacent to the 65° angle, so cos 65° = adjacent/20 and adjacent = 20 cos 65° ≈ 20(0.4226) ≈ 8.5 feet. The choice 18.1 is 20 sin 65°, the height up the wall, which uses the opposite side instead of the adjacent one. The two large values, 42.9 and 47.3, come from dividing by the trig ratio rather than multiplying, and both are impossible because no leg can be longer than the 20-foot hypotenuse. Tip: sketch the triangle and label the hypotenuse first. Any answer bigger than the hypotenuse can be eliminated before you touch the calculator.
Question 3 hard

In a 30°-60°-90° triangle, the leg opposite the 60° angle has length 6√3. What is the length of the hypotenuse?

  1. A. 6
  2. B. 6√3
  3. C. 12 correct
  4. D. 18
Why: In a 30°-60°-90° triangle the sides are in the ratio 1 : √3 : 2, with the longer leg equal to the shorter leg times √3. Here 6√3 is the longer leg, so the shorter leg is 6 and the hypotenuse is 2(6) = 12. The choice 18 multiplies the given leg by √3 again, treating the longer leg as if it were the shorter one, and 6 stops at the shorter leg. Tip: in special right triangles, identify which side you were handed before scaling the 1 : √3 : 2 ratio.

The explanation is the point. Getting a question wrong tells you nothing you did not already suspect; knowing which step you missed is what changes the next one.

Practise the rest of the 16

The other 13 questions on this skill are in the free question bank, filterable by difficulty, each with the same worked explanation. No account and no card.

Open the Math bank

Frequently asked questions

How many ACT right triangles & trigonometry questions does Prep36 have?
16, each tagged to this skill and each with a worked explanation.
Are these real ACT questions?
No. They are original questions written to the Enhanced ACT format. Prep36 is not affiliated with ACT, Inc.
Is the Math section part of the composite?
Yes. On the Enhanced ACT the composite is the average of English, Math and Reading. Science is optional and reported separately.
Keep readingACT Math practiceEvery Math skill in one place.How many questions can you miss?The miss budget for every score.ACT Score CalculatorTurn raw scores into a composite.
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