Question Video: Solving for the Hypotenuse of a Right Triangle Mathematics • 8th Grade

Find 𝑥 in the right triangle shown.

03:33

Video Transcript

Find 𝑥 in the right triangle shown.

Looking at the information we’ve been given, we note, first of all, that this triangle is a right triangle. It includes a right angle. And we’ve been given the lengths of two of its sides. They are eight units and 15 units. 𝑥 represents the length of the third side of this right triangle. And from its position, directly opposite the right angle, we note that 𝑥 is the hypotenuse of this triangle. As we’ve been given the lengths of two sides in a right triangle and we wish to calculate the third, this is exactly the setup we need in order to apply the Pythagorean theorem.

This tells us that, in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. So we’ll begin by writing down what the Pythagorean theorem tells us about this triangle in particular. The two shorter sides are eight units and 15 units. So the sum of the squares of the two shorter sides is eight squared plus 15 squared. This is then equal to the square of the hypotenuse. And as the hypotenuse of our triangle is 𝑥, we now have the equation eight squared plus 15 squared is equal to 𝑥 squared.

So by considering what the Pythagorean theorem tells us about this triangle in particular, we have an equation we can solve in order to determine the value of 𝑥. You may prefer to swap the two sides of the equation around so that 𝑥 is on the left-hand side, although this isn’t entirely necessary. Now that we formed our equation, we’re going to solve it by first evaluating eight squared and 15 squared. This gives 𝑥 squared equals 64 plus 225, which simplifies to 𝑥 squared equals 289.

The next step in solving this equation is to take the square root of each side because the square root of 𝑥 squared will give 𝑥. Now usually, when we solve an equation by square rooting, we must remember to take plus or minus the square root. But here 𝑥 has a physical meaning; it’s the length of a side in a triangle. So it must take a positive value. We therefore write 𝑥 equals just the positive square root of 289. 289 is in fact a square number, and its square root is 17. So we found the value of 𝑥. 𝑥 is equal to 17.

Now, we should always perform a quick sense check of our answer by comparing the value we found with the other two sides in the triangle. Remember, 𝑥 represents the hypotenuse, which is the longest side in this right triangle. So our value for 𝑥 needs to be bigger than the lengths of the two other sides. Our value is 17 and the two other sides are 15 and eight. So our answer does make sense.

Now, in fact, this triangle is an example of a special type of right triangle, called a Pythagorean triple. This is a right triangle in which all three of the side lengths are integers. The most well-known Pythagorean triple is the three-four-five triangle as three squared plus four squared is equal to five squared. You may well encounter Pythagorean triples when working without a calculator. So it’s a good idea to be familiar with some of the most common ones. By applying the Pythagorean theorem then, we’ve found the value of 𝑥 in the right triangle shown is 17.

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