Determine, to the nearest 10th, the perimeter of a right triangle, given that its hypotenuse is 49 centimeters and that one of its sides is 26 centimeters.
So here we have a right triangle. The hypotenuse side is the longest side. It’s the side across from the 90-degree angle, so we can label that 49 centimeters. We also know that one of its sides, so either of the two that are remaining, is 26 centimeters. So we can choose either one. So now that we’ve chosen the bottom side to be 26 centimeters, we can label the remaining side 𝑥 in centimeters. So what we can use to solve for this last missing side. That way we can find the perimeter because the perimeter is the sum of all of the sides. So we need to find that missing side length before we can do that, and we can use the Pythagorean theorem to do so.
The Pythagorean theorem states that we can square the longest side and set it equal to the sum of the squares of the shorter sides. So we have 𝑥 squared plus 26 squared equals 49 squared, because again, we can set the longest side squared equal to the sum of the squares of the shorter sides. So now let’s evaluate. 26 squared is 676, and 49 squared is 2401. So to solve for 𝑥 squared, we need to subtract 676 from both sides of the equation. So 𝑥 squared is equal to 1725. So now we need to square root both sides in order to get 𝑥 all alone. And the square root of 1725 is 41.53319.
Now when it said to find the perimeter, it said to round to the nearest 10th. So that would be here, one decimal place. So we need to look at the number to the right of that, which is a three. And since three is less than five, we will keep this five a five, we will not round up. So 𝑥 is equal to 41.5. So we can label that on our triangle, so that side length is 41.5 centimeters.
So to find the perimeter, we need to add all of the sides together. So 41.5 plus 26 plus 49, so our perimeter is 116.5 centimeters.