Is this triangle a right triangle?
A right triangle is a triangle in which one of the angles is a right angle. In order to determine whether this triangle is a right triangle, we need to consider whether or not the Pythagorean theorem holds as the Pythagorean is only true in right triangles.
The Pythagorean theorem tells us that in a right triangle the square of the hypotenuse, which is the longest side, is equal to the sum of the squares of the two shorter sides. If the two shorter sides are labelled as 𝑎 and 𝑏 and the hypotenuse is labelled as 𝑐, then this means that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared. Let’s consider whether this relationship is true in the triangle that we’ve been given.
The two shorter sides of the triangle are five centimeters and 12 centimeters. So the sum of their squares is five squared plus 12 squared. This is equal to 25 plus 144 which is 169. The longest side of the triangle is 13 centimeters. So its square is equal to 13 squared. This is equal to 169.
As the sum of the squares of the two shorter sides is equal to the square of the longest side, the Pythagorean theorem holds for this triangle. And therefore, it is a right triangle. In fact, the numbers five, 12, and 13 are an example of a Pythagorean triple that is a right triangle, in which the lengths of all three sides are integers.