We are asked to explain how we can determine whether a triangle is a right triangle. Given a triangle, we could draw it and measure the angles. However, it would be very difficult to draw the triangle and measure the angles precisely, so this is not a perfect way to determine whether a triangle is a right triangle. Instead, we can use the Converse of the Pythagorean Theorem.

Converse of the Pythagorean Theorem
If the sum of squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, the triangle is a right triangle.

Let's also recall the Pythagorean Theorem.

The Pythagorean Theorem
In a right triangle, the sum of squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

If the sum of the squares of the lengths of the two shorter sides of the triangle is equal to the square of the length of the longest side, the triangle is a right triangle by the Converse of the Pythagorean Theorem. If that is not the case, then by the Pythagorean Theorem, the triangle cannot be a right triangle. Let's consider an example triangle.

Here, $c$ is the longest side. If the sum of $a^{2}$ and $b^{2}$ is equal to $c^{2},$ then $△ A B C$ is a right triangle. If $a^{2} + b^{2}$ is different from $c^{2},$ then $△ A B C$ is not a right triangle. In general, in a triangle with side lengths $a,$ $b,$ and $c,$ where $c$ is the length of the longest side, the triangle is a right triangle if and only if $a^{2} + b^{2}$ is equal to $c^{2} .$

Exercise