Find 𝑥 in the right triangle.
Angle 𝐷𝐹𝐸 is a right angle. We’re missing the value of length 𝐷𝐸, which is opposite the right angle. That makes 𝑥 the hypotenuse. We’re not given any information about the interior angles. But we do know the values of the other two side lengths. And that means we can use the Pythagorean theorem, which tells us the square sum of the two smaller side lengths equals the side length 𝑐, the hypotenuse, squared.
For us, that means five squared plus 12 squared equals 𝑥 squared. 25 plus 144 equals 𝑥 squared. 169 equals 𝑥 squared. We take the square root of both sides. And the square root of 169 has two solutions, positive 13 and negative 13. But because we’re interested in side lengths, we’re only interested in the positive solution. 𝑥 must equal 13.
Now, before we move on, we have side lengths five, 12, and 13. This is a special group of side lengths that we call a Pythagorean triple. It’s called that because each of the values of 𝑎, 𝑏, and 𝑐 are positive integers. They’re whole numbers.
Another common Pythagorean triple is three, four, and five. When you know the Pythagorean triple five, 12, 13, you can immediately see the five and the 12 inside the right triangle as the two smallest sides and recognise that the hypotenuse would be 13 because it is a Pythagorean triple. If you don’t know this set as a Pythagorean triple, then you can you use Pythagorean theorem to solve.
Either way, 𝑥 is equal to 13.