Pythagorean theorem & right triangles drills

Use the Pythagorean theorem to find side lengths and distances in right triangles.

Free, timed SHSAT practice sets on pythagorean theorem & right triangles— every question includes a full worked explanation, and there's no account required.

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How to solve Pythagorean theorem problems

In a right triangle, \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse: the longest side, across from the right angle.

  1. Find the hypotenuse first, so you know which side is \(c\).
  2. Missing hypotenuse: add the squares, then take the square root. Legs 8 and 15: \(\sqrt{64 + 225} = \sqrt{289} = 17\).
  3. Missing leg: subtract. Hypotenuse 10, leg 6: \(\sqrt{100 - 36} = \sqrt{64} = 8\).
  4. If the answer is not a whole number, simplify: legs 3 and 6 give \(\sqrt{45} = 3\sqrt{5}\).

In word problems, look for the right angle: a ladder against a wall, or the diagonal of a rectangle. Knowing the triples 3-4-5, 5-12-13, 8-15-17, 7-24-25 saves time.

Common traps: forgetting the final square root, adding the legs (\(8 + 15 = 23\) is not the hypotenuse), and adding squares when you need a leg.

Worked examples

Example 1.
A right triangle has hypotenuse 15 m and one leg 12 m. What is the length of the other leg?
Show answer and explanation
Answer:
D

By the Pythagorean theorem, the other leg \(= \sqrt{15^2 - 12^2} = \sqrt{81} = 9\) m. (Stopping at \(15^2 - 12^2 = 81\) forgets the square root. Adding the squares, as if 15 were a leg, gives \(3\sqrt{41}\). Subtracting the lengths without squaring, \(15 - 12\), gives 3.)

Example 2.
A right triangle has legs 7 m and 8 m. What is the exact length of the hypotenuse?
Show answer and explanation
Answer:
C

By the Pythagorean theorem, \(c = \sqrt{7^2 + 8^2} = \sqrt{113}\) m. (Stopping at \(7^2 + 8^2 = 113\) forgets the square root. Adding the legs, \(7 + 8\), gives 15. Subtracting the squares, \(\sqrt{8^2 - 7^2}\), gives \(\sqrt{15}\).)

Example 3.
A rectangular field is 3 ft long and 4 ft wide. What is the straight-line distance from one corner to the opposite corner?
Show answer and explanation
Answer:
C

This is a right triangle with legs 3 ft and 4 ft, so the missing length \(= \sqrt{3^2 + 4^2} = \sqrt{25} = 5\) ft. (Stopping at \(3^2 + 4^2 = 25\) forgets the square root. Adding the legs, \(3 + 4\), gives 7. Subtracting the squares, \(\sqrt{4^2 - 3^2}\), gives \(\sqrt{7}\).)