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Digital SAT · Right-Triangle Trigonometry

How to Solve Right Triangles and the Pythagorean Theorem on the Digital SAT

In a right triangle, the two legs squared add up to the hypotenuse squared: a squared plus b squared equals c squared. Sort out which sides are legs and which is the hypotenuse, then solve for the one you are missing. Knowing a few triples, like 3-4-5 and 5-12-13, skips the arithmetic. When triangles are similar, their sides are proportional, so a ratio does the work. Desmos can take the squares and the square root.

Written from Perfect1600’s analysis of every right triangles and the pythagorean theorem question in our bank·Method checked against the current Bluebook test
per test
less than 1 per test
per test
typical difficulty
Medium to hard
typical difficulty
practice questions
114
practice questions
How we counted

Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.

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What the question bank shows

a²+b²=c²
Legs squared add up

Name the hypotenuse first, then solve; common triples skip the arithmetic.

75%
Do it in Desmos

Type the square-root expression and Desmos returns the exact side.

the hazard
Hypotenuse placement

Putting a leg where the hypotenuse belongs flips the equation.

How to recognize right triangles and the pythagorean theorem questions

  • A right triangle is shown or described, two sides known and one missing.
  • The question asks for a leg, the hypotenuse, or a distance forming a right triangle.
  • The numbers hint at a triple, like 3-4-5, 5-12-13, or 8-15-17.
  • Similar or proportional triangles appear, so corresponding sides form a ratio.

Why students miss these

The theorem is short, but the hypotenuse gets misplaced and similarity gets mismatched. The hypotenuse is always the longest side, across from the right angle, and putting a leg there flips the equation. People also add the sides instead of their squares, or forget the final square root. With similar triangles, pairing sides that do not correspond gives a plausible wrong length. Name the hypotenuse first and the setup holds.

The step-by-step method

  1. 1

    Find the hypotenuse

    It is the longest side, across from the right angle; the other two are legs.

  2. 2

    Apply the theorem

    Legs squared sum to the hypotenuse squared: a squared plus b squared equals c squared. Solve for the missing side.

  3. 3

    Use triples or similarity

    Recognize common triples to skip arithmetic, and for similar triangles pair corresponding sides in a proportion.

  4. 4

    Take the square root

    Solve for the side and take the root, then report the length asked.

Solving it on Desmos

  1. Type the theorem. Enter sqrt(9^2 + 12^2) for a hypotenuse, and read the value.
  2. Solve for a leg. Type sqrt(c^2 - a^2) with the known hypotenuse and leg.
  3. Compute a proportion. For similar triangles, type the cross-multiplied ratio and read the missing side.

Full Desmos walkthrough for right triangles and the pythagorean theorem→

When to use it: Desmos takes the squares and the root cleanly, sparing you the arithmetic. Naming the hypotenuse and matching corresponding sides in a similarity are the calls it leaves to you.
See it live: open a real question in the same Desmos calculator you get on Perfect1600, with the equations already typed in.

Worked examples

Easy example
A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?
13
B
14
C
15
D
17

A: Correct. c=52+122=169=13c = \sqrt{5^{2} + 12^{2}} = \sqrt{169} = 13.

B: Incorrect. This does not satisfy the Pythagorean theorem.

C: Incorrect. This does not satisfy 52+122=c25^{2} + 12^{2} = c^{2}.

D: Incorrect. This adds the legs instead of squaring them.

Explanation

c=25+144=169=13c = \sqrt{25 + 144} = \sqrt{169} = 13.

Easy example
A right triangle has legs of length 9 and 12 and a hypotenuse of length cc. Which equation correctly relates the side lengths?
A
9+12=c9 + 12 = c
92+122=c29^{2} + 12^{2} = c^{2}
C
92−122=c29^{2} - 12^{2} = c^{2}
D
9+12=c\sqrt{9} + \sqrt{12} = c

A: Incorrect. The legs are not simply added to give the hypotenuse.

B: Correct. The Pythagorean theorem states a2+b2=c2a^{2} + b^{2} = c^{2} for the legs and hypotenuse.

C: Incorrect. The legs are added, not subtracted, under the Pythagorean theorem.

D: Incorrect. The square roots of the legs are not summed to give the hypotenuse.

Explanation

For a right triangle, 92+122=c29^{2} + 12^{2} = c^{2}.

Medium example
A right triangle has a hypotenuse of length 25 and one leg of length 7. What is the length of the other leg?
A
14
B
18
C
20
24

A: Incorrect. This does not satisfy 72+b2=2527^{2} + b^{2} = 25^{2}.

B: Incorrect. This subtracts the lengths directly instead of squaring.

C: Incorrect. This does not satisfy the Pythagorean theorem.

D: Correct. b=252−72=576=24b = \sqrt{25^{2} - 7^{2}} = \sqrt{576} = 24.

Explanation

b=625−49=576=24b = \sqrt{625 - 49} = \sqrt{576} = 24.

The common traps

PatternWhat it doesThe tell
Hypotenuse in the wrong placeTreated a leg as the hypotenuse, flipping the equation.The hypotenuse is the longest side, across from the right angle.
Added sides, not squaresAdded the legs directly instead of their squares.The theorem adds the squares of the legs, then takes a root.
Forgot the square rootReported c squared instead of c.The last step is a square root to get the side.
Mismatched similarityPaired non-corresponding sides in a proportion.Match corresponding sides by position before writing the ratio.

Try it: two real questions

Question 1easy

A right triangle has legs of length 3 and 4. What is the length of the hypotenuse?

Question 2easy

A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

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Often confused with

Lines, Angles & Triangles
Angle Relationships
Area & Volume
Area and Perimeter
Right-Triangle Trigonometry
Trigonometric Ratios (SOH-CAH-TOA)

Related reading

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Common questions

How do you use the Pythagorean theorem on the SAT?

For a right triangle, add the squares of the legs to get the square of the hypotenuse: a squared plus b squared equals c squared. Solve and take the square root. In Desmos, type the square-root expression.

What are the common Pythagorean triples?

3-4-5, 5-12-13, 8-15-17, and their multiples like 6-8-10. Spotting one lets you write the missing side with no calculation.

How do I know which side is the hypotenuse?

It is the longest side, across from the right angle. The two shorter sides are the legs, and their squares add.

How do similar triangles help find a side?

Their corresponding sides are proportional, so set up a ratio pairing sides in the same positions and solve. Make sure the sides you pair correspond.

Can Desmos compute right-triangle sides?

Yes. Type sqrt(5^2 + 12^2) for a hypotenuse or sqrt(c^2 - a^2) for a leg, and read the value. It avoids squaring and square-root slips.

Practice right triangles and the pythagorean theorem the way it is tested

Start with the free 16-question diagnostic, then drill this type with step-by-step reasoning on every question.

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