Video: Solving for One of the Legs of a Right Triangle

Find 𝑥 in the right triangle shown.

03:14

Video Transcript

Find 𝑥 in the right triangle shown.

So we have a right triangle. And we’re asked to find the value of 𝑥, which represents the length of one of the triangle sides. We’ve been given the lengths of the other two sides. So we have exactly the right set of information 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. Now, before applying the Pythagorean theorem, we must be very careful to make sure we correctly identify which side of the triangle is the hypotenuse. And remember, it’s always the side directly opposite the right angle. So in this case, the hypotenuse of the triangle is 13 units.

The side we’ve been asked to find, length 𝑥, is one of the two shorter sides or legs of this right triangle. So the first thing we do then is to write down what the Pythagorean theorem tells us about this particular triangle. The two shorter sides are 𝑥 and 12. So the sum of their squares will be 𝑥 squared plus 12 squared. The hypotenuse of the triangle is 13 units. So if the sum of the squares of the two shorter sides is equal to the square of the hypotenuse, we have the equation 𝑥 squared plus 12 squared equals 13 squared.

Now that we’ve formed our equation, we’re going to solve it to determine the value of 𝑥. First, we evaluate 12 squared and 13 squared, giving 𝑥 squared plus 144 equals 169. We want to leave 𝑥 or 𝑥 squared initially on its own on the left-hand side of the equation. So the next step is to subtract 144 from each side. On the left-hand side, 𝑥 squared plus 144 minus 144 just leaves 𝑥 squared. And on the right-hand side, 169 minus 144 is 25.

The final step is to take the square root of each side of the equation, remembering we only need to take the positive square root as 𝑥 represents a length. So it must have a positive value. 𝑥 is therefore equal to the square root of 25. And as 25 is a square number, its square root is an integer; it’s simply five. So we found the value of 𝑥. 𝑥 is equal to five. Now, in fact, this triangle is an example of a Pythagorean triple. That’s a right triangle in which all three side lengths are integers.

We should also perform a quick check of our answer. Remember, we were looking to calculate one of the shorter sides of this triangle. So our value for 𝑥 must be less than the length we were given for the hypotenuse. Five is certainly less than 13. So our answer makes sense. So by applying the Pythagorean theorem, we’ve solved this problem. The value of 𝑥 is five. We must make sure we’re really careful when setting up our equation. And we need to be sure before we begin whether we’ve been asked to find the length of one of the shorter sides or the length of the hypotenuse.

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