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Exercises on Pythagorean theorem in right triangles

Apply the Pythagorean theorem to calculate the hypotenuse or a leg, and check if a triangle is a right triangle. Theory and step-by-step solved exercises, with practice corrected instantly.

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Pythagorean theorem: quick theory

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the squares of the legs:

a2 + b2 = c2

To find the hypotenuse: c = √(a2 + b2). To find a leg: a = √(c2 − b2). It is also used to check if a triangle is right-angled.

Step-by-step solved exercises on Pythagorean theorem

Solved examples to show you the method. Then, practice with instant correction.

Exercise 1The legs of a right triangle measure 3 and 4. How long is the hypotenuse?

Step-by-step solution

  1. We apply the theorem: c² = 3² + 4² = 9 + 16 = 25.
  2. We take the root: c = √25.
Result: the hypotenuse is 5.

Exercise 2The hypotenuse measures 13 and one leg measures 5. How long is the other leg?

Step-by-step solution

  1. We isolate one leg: a² = c² − b² = 13² − 5² = 169 − 25 = 144.
  2. We take the root: a = √144.
Result: the other leg is 12.

Exercise 3Is a triangle with sides 6, 8, and 10 a right triangle?

Step-by-step solution

  1. We check if the theorem holds with hypotenuse = longest side (10).
  2. 6² + 8² = 36 + 64 = 100, and 10² = 100.
  3. Since 100 = 100, it holds true.
Result: yes, it is a right triangle.

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This topic is studied in:

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Frequently asked questions about Pythagorean theorem

What is the Pythagorean theorem used for?

It allows you to calculate one side of a right triangle if you know the other two, and to check if a triangle is a right triangle. It is widely used in geometry and distance problems.

How do you calculate a leg with the Pythagorean theorem?

Solve for: leg = √(hypotenuse² − other leg²). That is, subtract the square of the known leg from the square of the hypotenuse and take the square root.

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