Problem 5. If one side of the square is extended by $2 \mathrm{~cm}$, and its adjacent side by $5 \mathrm{~cm}$, then we get a rectangle whose area is $45 \mathrm{~cm}^{2}$ larger than the area of the square.

What is the area of the square?
Solution. Let the side of the square have length $a$. The rectangle obtained by extending the sides of the square has sides $a+2$ and $a+5$. The area of the square is $a \cdot a$, and the area of the rectangle is $(a+2) \cdot (a+5)$. However, the rectangle is actually obtained by drawing three rectangles around the given square, with areas $2a, 5a, 2 \cdot 5$ (see diagram). Then the total increase in the area of the square is

$$
45 \mathrm{~cm}^{2}=2a+5a+10=(7a+10) \mathrm{~cm}^{2}
$$

From this, $7a=35$, so the side of the square is $a=5 \mathrm{~cm}$.

The area of the square is $25 \mathrm{~cm}^{2}$.

## VI section