4. The diagram shows two symmetrically placed squares with sides of length 2 and 5 .
What is the ratio of the area of the small square to that of the shaded region?
A $7: 24$
B $1: 3$
C $8: 25$
D $8: 21$
E $2: 5$
Solution
D

The small square has area $2^{2}$, that is, 4 . The shaded region is made up of half the large square with side length 5 , with half the small square with side length 2 removed.
Therefore, the shaded area is $\frac{1}{2}\left(5^{2}\right)-\frac{1}{2}\left(2^{2}\right)=\frac{25}{2}-2=\frac{21}{2}$. Hence the ratio of the area of the small square to that of the shaded region is $4: \frac{21}{2}$, that is, $8: 21$.

For INVESTIGATION
4.1 Suppose that the area of the shaded region were one-third of the area of the outer square. What would be the ratio of the area of the inner square to the area of the outer square?
4.2 Suppose that the ratio of the shaded area to the area of the small square is $7: 18$. What is the ratio of the side length of the small square to the side length of the large square?