Determine the diagonal length of the rectangle whose perimeter is 326.4 centimetres and length 115.2 centimetres.
So what I’ve done here is I’ve drawn a sketch. And we’ve got the information that we’ve been given in the question. So we’ve got our rectangle. It’s got a length 115.2 centimetres, and its perimeter is 326.4 centimetres. Now, what else can I mark onto our diagram? Well, I can mark the opposite side to the one that I’ve marked already because the length of the rectangle is gonna be the same on the top and the bottom. So we’re gonna have 115.2 centimetres. And I’m also gonna mark on our width. And our width is gonna be 𝑥 on each side. And that’s because it’s gonna be the same. I’ve decided to call it 𝑥 because that’s gonna be useful in a minute when we want to try and find it. And I’ve also marked on our diagonal because this is what we’re looking to find.
Well, if we want to find the diagonal, we can see that we’ve got a right-angled triangle, if we split our rectangle in two. And this right-angled triangle is gonna have 𝑑, so the diagonal of our rectangle, as its hypotenuse. So therefore, as we have a right-angled triangle and we want to find the hypotenuse, we can utilize the Pythagorean theorem. And that is that 𝑎 squared plus 𝑏 squared equals 𝑐 squared. So if we square each of these shorter sides and add them together is equal to the longest side or our hypotenuse squared. But there’s only one problem. We don’t yet know what our shorter side is. So we’re gonna have to calculate this. And we can do this using the fact that we know the perimeter of our rectangle. And that’s because the perimeter of our rectangle is the distance all the way round.
So therefore, we know the perimeter is gonna be the length add the width add the length add the width. So the perimeter is gonna be equal to 115.2 plus 𝑥 plus 115.2 plus 𝑥. But we know what the perimeter is equal to because we’re told it’s 326.4. So what we can do is we can set it into an equation. So we can say that 326.4 is equal to, and then what we had before. So now, what we need to do is simplify.
So first of all, what we could do is add our 𝑥s because we’ve got positive 𝑥 add 𝑥, which is just two 𝑥. And then, we have 115.2 add 115.2, which is 230.4. So then, what we can do is subtract 230.4 from each side of the equation. And when we do this to each side, we’re gonna get 96 is equal to two 𝑥. And that’s because if you take 230.4 away from 326.4, we get 96. And if you take it away from herself, we’re left with zero. So we’re just left with two 𝑥 on the right-hand side. And then, if we divide both sides by two, we’re left with 48 is equal to 𝑥. So we’ve found our value of 𝑥.
So now, as we found 𝑥, so we know the width of our rectangle, we can apply the Pythagorean theorem to find our diagonal. And that’s because we can look at the right-angled triangle that we’ve formed. And we can see that our diagonal is our hypotenuse. So therefore, we can see that our diagonal squared is equal to 115.2 squared plus 48 squared. So therefore, our diagonal squared is equal to 13271.04 plus 2304. So our diagonal squared is equal to 15575.04.
So now, there’s one more thing to do. And that is to find our 𝑑, so our diagonal. Don’t leave the answer as this because it’s an often a common mistake because we’ve got here 𝑑 squared. So therefore, if we want to find 𝑑, so our diagonal, what we need to do is take the square root of both sides of our equation. And when we do that, we get 𝑑 is equal to 124.8. So therefore, we can say that the diagonal length of a rectangle whose perimeter is 326.4 centimetres and length 115.2 centimetres is 124.8 centimetres.