The Pythagorean theorem states:
In a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse.
The converse of the Pythagorean theorem states:
If in a triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse, then the triangle is a right triangle.
We need to show that the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse.
One leg has length 3. Its square is 9.
The other leg has length 4. Its square is 16.
The sum of the squares is 9 + 16 = 25.
The hypotenuse has length 5. Its square is 25.
The sum of the squares of the lengths of the legs is indeed equal to the square of the length of the hypotenuse, so this triangle is a right triangle.
