Video: Solving for the Hypotenuse of a Right Triangle

Determine the diagonal length of the rectangle whose length is 48 cm, and width is 20 cm.

03:29

Video Transcript

Determine the diagonal length of the rectangle whose length is 48 centimetres and width is 20 centimetres.

Now, we haven’t been given a diagram for this question. So it’s always a good idea to begin by drawing our own. We have a rectangle with a length of 48 centimetres and a width of 20 centimetres. The length we’ve been asked to calculate is the diagonal of this rectangle. That’s the line that joins opposite corners together. We can use the letter 𝑑 to represent this unknown length. Now we know that all the interior angles in a rectangle are 90 degrees. So, in fact, this problem isn’t just about rectangles. It’s also about right triangles, that is, the triangle formed by the rectangle’s length, its width, and this diagonal.

Looking at the lower triangle in our diagram, we can see that we’ve been given the lengths of two of its sides — they’re 20 centimetres and 48 centimetres — and asked to calculate the length of its third side. And as this is a right triangle, we’re going to be able to do this by applying 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 we try to apply the Pythagorean theorem, we must identify which of the three sides we’ve been asked to calculate. Remember, the hypotenuse is always the side directly opposite the right angle. So the side we’re looking to find is the hypotenuse of a triangle.

We then ask ourselves, “what does the Pythagorean theorem tell us, not just in general, but about this triangle specifically?” Well, as the two shorter sides are 48 and 20 centimetres, the sum of their squares is 48 squared plus 20 squared. And the square of the hypotenuse is 𝑑 squared. So we have the equation 48 squared plus 20 squared equals 𝑑 squared. We can of course swap the two sides of the equation round if we prefer to have 𝑑 squared on the left-hand side. So by applying the Pythagorean theorem, we’ve formed an equation, which we can now solve in order to determine the value of 𝑑.

First, we evaluate 48 squared and 20 squared and then add these values together to give 𝑑 squared is equal to 2704. The final step in solving this equation is to take the square root of each side, giving 𝑑 equals the square root of 2704. Now, 2704 is in fact a square number although probably not one that you are overly familiar with. Its square root is simply 52. So we have that 𝑑 is equal to 52. The diagonal length of this rectangle then is 52 centimetres.

Now, we should perform a quick sense check of our answer. Remember, 𝑑 was the hypotenuse of this triangle. It’s supposed to be the longest side. So we need to check that our value does make sense. Well, 52 is indeed greater than each of the other side lengths. So it’s a sensible value for the hypotenuse of this triangle. So we’ve completed the problem. The key stage in this question was to first draw our own diagram. And once we did, we saw that this problem wasn’t just about rectangles. It was in fact about right triangles. And hence, we could solve it by applying the Pythagorean theorem.

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