24. The diagram shows a regular octagon with sides of length 1 . The octagon is divided into regions by four diagonals.
What is the difference between the area of the hatched region and the area of the region shaded grey?
A 0
B $\frac{1}{8}$
C $\frac{1}{4}$
D $\frac{1}{2}$
E 1
Solution
C The octagon is divided into the central square, which has sides of length 1 , four congruent rectangles and four congruent right-angled triangles. The hatched region is made up of two of the rectangles, and the square. The grey area is made up of two of the rectangles and three of the triangles. Therefore the difference between their areas is the difference between the area of the square and the area of three of the triangles.

The square has area 1. Each of the triangles is a right-angled isosceles triangle with hypotenuse of length 1 . Let $x$ be the length of each of the other two sides. By Pythagoras' Theorem, $x^{2}+x^{2}=1^{2}$, that is $2 x^{2}=1$ and so $x^{2}=\frac{1}{2}$. So the area of each triangle is $\frac{1}{2} x^{2}=\frac{1}{4}$. Therefore the difference between the area of the square and the area of three of the triangles is $1-3\left(\frac{1}{4}\right)=\frac{1}{4}$.
We can also see this geometrically. The diagram on the right shows that four of the triangles fit together to make a square whose side has length 1. So the difference between the area of the square and that of three of the triangles is the area of one-quarter of the square.