Here is a right-angled
triangle. Four such triangles are placed as
shown in the diagram to form the square 𝐴𝐵𝐶𝐷. Find the area of the square
𝐴𝐵𝐶𝐷 in terms of 𝑥 and 𝑦.
There are several elements to this
question. Remember, the area of a rectangle
is found by multiplying its width by its height. In the case of a square, since its
width and its height are equal, it’s the same as finding its width and multiplying
it by itself, or squaring it. So how do we find the width of the
Well, the trick is to remember that
these are all right-angled triangles. Each side of the square is made up
of the longest side, the hypotenuse, of a triangle. We’ve been given the lengths of the
other two sides of this triangle. So we can use Pythagoras’s theorem
to help us find an expression for the length of the hypotenuse.
Remember, 𝑐 is always the
hypotenuse. In this case, that’s what we’re
trying to find. It’s the length 𝐴𝐵, which is the
width of the square. That means we can rewrite
Pythagoras’s theorem as 𝑥 squared plus 𝑦 squared is equal to 𝐴𝐵 squared.
Usually, we would then find the
square root of both sides of this equation to find the length of the hypotenuse. But remember, we said that, to find
the area of the square, we can take the width and square it. However, the width is actually
𝐴𝐵. And when we square that, we get
𝐴𝐵 squared. So the area of the square is given
by 𝐴𝐵 squared. But we already said that that was
the same as 𝑥 squared plus 𝑦 squared. The expression for the area of the
square is, therefore, 𝑥 squared plus 𝑦 squared.