Geometry and Trigonometry · Math
Right triangles and trigonometry
Right triangles and trigonometry: the Pythagorean theorem, the special 45-45-90 and 30-60-90 ratios, and sine, cosine, and tangent.
20 questions in the bank · 3 easier · 9 medium · 8 harder · this domain is ≈15% of the section
What to watch for
Identify which side is opposite the angle in question, not merely which is shortest. Sine of an angle equals cosine of its complement, which is why those questions pair up.
How the question is worded
These are the actual stems used in the bank — the SAT reuses a small set of wordings, and recognising them saves time on test day.
- The legs of a right triangle have lengths 7 and 24. What is the length of the hypotenuse?
- A right triangle has a hypotenuse of length 15 and one leg of length 9. What is the length of the other leg?
- The legs of a right triangle have lengths 10 and 24. What is the length of the hypotenuse?
- A right triangle has a hypotenuse of length 20 and one leg of length 12. What is the length of the other leg?
Worked examples
easyThe legs of a right triangle have lengths 7 and 24. What is the length of the …
The legs of a right triangle have lengths 7 and 24. What is the length of the hypotenuse?
Answer: 25
Explanation
hypotenuse² = 7² + 24² = 625, and √625 = 25. This is the 7-24-25 right triangle — one of the triples worth recognizing on sight.
mediumA right triangle has a hypotenuse of length 15 and one leg of length 9. What i…
A right triangle has a hypotenuse of length 15 and one leg of length 9. What is the length of the other leg?
Answer: 12
Explanation
By the Pythagorean theorem, leg² = 15² − 9² = 225 − 81 = 144, so the leg is 12. Subtracting (not adding) when the hypotenuse is known is the key step.
hardAn isosceles right triangle has two legs of length 4. What is the length of th…
An isosceles right triangle has two legs of length 4. What is the length of the hypotenuse?
- A. 4√2
- B. 8
- C. 4
- D. 4√3
Answer: A. 4√2
Explanation
In a 45°−45°−90° triangle the hypotenuse is leg·√2: 4√2. From the Pythagorean theorem: √(4² + 4²) = √32 = 4√2 — the ratio 1 : 1 : √2 is worth memorizing.