Calculate the perimeter of a rhombus whose two diagonals have lengths of 56 centimeters and 90 centimeters.
When we begin a question like this, it can be helpful to draw a diagram so that we can clearly visualize the information that we’re given. So, here we have our rhombus. This is a quadrilateral that has all four sides the same length. We have two diagonals on our rhombus. The longer one here will be the 90-centimeter one, and the shorter one will be 56 centimeters. In this question, we need to calculate the perimeter. So, let’s start by working out the length of one of the sides and we can call this 𝑥.
In this question, we need to know a key fact about the diagonals of a rhombus. And, that is that the diagonals of a rhombus are perpendicular bisectors. The word perpendicular means that they cross at 90 degrees. And when they bisect, that means that they cut each other exactly into two pieces of equal length.
So, let’s take a closer look at one of the triangles that is formed by the diagonals. Since the diagonals are perpendicular bisectors, the angle of this triangle will be 90 degrees. The length of the base of the triangle will be 45 centimeters because that’s exactly half of 90 centimeters. And, the length on the height of the triangle will be 28 centimeters, since that’s exactly half of 56 centimeters. And, the length on the hypotenuse of our triangle will be 𝑥, which was the original side length of our rhombus.
In order to find our unknown value 𝑥, we can use Pythagoras’s theorem since this is a right-angle triangle. Pythagoras’s theorem says that the length of the hypotenuse squared is equal to the sum of the squares on the other two sides. So substituting in our values for, say, the hypotenuse and the other two sides, 𝑎 and 𝑏, will give us 𝑥 squared equals 28 squared plus 45 squared. And, it doesn’t matter which way around we have our 28 and our 45. Using our calculator, we can evaluate this to give us 𝑥 squared equals 784 plus 2025. Adding the values on the right-hand side of our equation will give us 𝑥 squared equals 2809.
In order to find 𝑥 on its own, we take the square root of both sides. And, using our calculator again to evaluate the root of 2809 will give us 𝑥 equals 53 centimeters. And so, we have calculated the side length of our rhombus will be 53 centimeters. So finally, in order to work out the perimeter, we add 53 plus 53 plus 53 plus 53. That’s the same as four lots of 53. Giving us our final answer for the perimeter of our rhombus as 212 centimeters.