Pythagorean Theorem Practice Problems

What this problem type covers

Each problem gives two sides of a right triangle and asks for the third. The first bands use Pythagorean triples so the answer is a whole number; later bands leave an irrational answer that has to be written as a simplified radical.

Example problems

One example from each difficulty band, with worked solutions.

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1. 1 · ★1
   A right triangle has legs of 60 and 63. Find the hypotenuse.
   • ①c = 123
   • ②c = 60
   • ③c = 7569
   • ④c = 3√41
   • ⑤c = 87
   Show solution

   ⑤c = 87

   1. Use the Pythagorean theorem: a² + b² = c²
   2. Substitute the known sides: 60² + 63² = c²
   3. So the unknown side squared is 3600 + 3969 = 7569.
   4. Taking the positive square root: c = 87
2. 2 · ★2
   A right triangle has legs of 12 and 35. Find the hypotenuse.
   • ①c = 1369
   • ②c = 37
   • ③c = 12
   • ④c = 47
   • ⑤c = √1081
   Show solution

   ②c = 37

   1. Use the Pythagorean theorem: a² + b² = c²
   2. Substitute the known sides: 12² + 35² = c²
   3. So the unknown side squared is 144 + 1225 = 1369.
   4. Taking the positive square root: c = 37
3. 3 · ★3
   A right triangle has a hypotenuse of 50 and one leg of 48. Find the other leg.
   • ①b = 2
   • ②b = 98
   • ③b = 196
   • ④b = 14
   • ⑤b = 2√1201
   Show solution

   ④b = 14

   1. Use the Pythagorean theorem: a² + b² = c²
   2. Substitute the known sides: 48² + b² = 50²
   3. So the unknown side squared is 2500 − 2304 = 196.
   4. Taking the positive square root: b = 14
4. 4 · ★4
   A right triangle has legs of 4 and 2. Find the hypotenuse.
   • ①c = 20
   • ②c = 2√5
   • ③c = 6
   • ④c = 4
   • ⑤c = 2√3
   Show solution

   ②c = 2√5

   1. Use the Pythagorean theorem: a² + b² = c²
   2. Substitute the known sides: 4² + 2² = c²
   3. So the unknown side squared is 16 + 4 = 20.
   4. Taking the positive square root: c = 2√5
5. 5 · ★5
   A right triangle has legs of 5 and 3. Find the hypotenuse.
   • ①c = 34
   • ②c = 8
   • ③c = 4
   • ④c = 5
   • ⑤c = √34
   Show solution

   ⑤c = √34

   1. Use the Pythagorean theorem: a² + b² = c²
   2. Substitute the known sides: 5² + 3² = c²
   3. So the unknown side squared is 25 + 9 = 34.
   4. Taking the positive square root: c = √34

Where people usually go wrong

• Treating the hypotenuse as just another leg. a² + b² = c² only holds with c as the hypotenuse — the longest side.
• Adding when you should subtract. Finding a leg means c² − a², not c² + a².
• Forgetting the square root at the end. c² = 169 is not the answer; c = 13 is.
• Rounding √50 to 7.07. The exact answer is 5√2, and rounding early loses it for good.
• Leaving a radical unsimplified. √72 and 6√2 are the same number, but only one is finished.