12. The diagram shows a 16 metre by 16 metre wall. Three grey squares are painted on the wall as shown.

The two smaller grey squares are equal in size and each makes an angle of $45^{\circ}$ with the edge of the wall. The grey squares cover a total area of $B$ metres squared.
What is the value of $B$ ?
SolUtion
128

The wall has a width of 16 metres so the diagonal of each smaller grey square is 8 metres. Let the side-length of each smaller grey square be $x$ metres. Then, by Pythagoras' Theorem, $x^{2}+x^{2}=8^{2}$, giving $x=\sqrt{32}$. Therefore each smaller grey square has an area of $32 \mathrm{~m}^{2}$.

The side-length of the larger grey square is equal to the length of the diagonal of one of the smaller grey squares. Therefore the larger grey square has area $8^{2}=64 \mathrm{~m}^{2}$.
Hence the total area covered by the grey squares, $B$, is $32+32+64=128 \mathrm{~m}^{2}$.