13. The pattern on a large square tile consists of eight congruent rightangled triangles and a small square. The area of the tile is $49 \mathrm{~cm}^{2}$ and the length of the hypotenuse $P Q$ of one of the triangles is $5 \mathrm{~cm}$. What is the area of the small square?
A $1 \mathrm{~cm}^{2}$
B $4 \mathrm{~cm}^{2}$
C $9 \mathrm{~cm}^{2}$
D $16 \mathrm{~cm}^{2}$
E $25 \mathrm{~cm}^{2}$
SolUtion
Since the four rectangles are congruent, the diagonal $P Q$ is also the side of a square. This square has area $(5 \times 5) \mathrm{cm}^{2}=25 \mathrm{~cm}^{2}$. Therefore the total area of the rectangles outside the square with side $P Q$ but inside the large square is $(49-25) \mathrm{cm}^{2}=24 \mathrm{~cm}^{2}$. However, this is also equal to the total area of the triangles inside the square with side $P Q$. Therefore the area of the small square is $(25-24) \mathrm{cm}^{2}=1 \mathrm{~cm}^{2}$.