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.
