14. The diagram shows a square, $K L M N$. A second square $P Q R S$ is drawn inside it, as shown in the diagram, where $P$ divides the side $K L$ in the ratio $1: 2$. Similarly, a third square $T U V W$ is drawn inside $P Q R S$ with $T$ dividing $P Q$ in the ratio $1: 2$.
What fraction of the area of $K L M N$ is shaded?
A $\frac{25}{81}$
B $\frac{16}{49}$
C $\frac{4}{9}$
D $\frac{40}{81}$
E $\frac{2}{3}$
Solution
A
Since $P$ divides the side $K L$ in the ratio $1: 2$, we choose units so that $K L$ has length 3 . Hence $K P$ has length 1 , and $P L$ has length 2 . We note also that it follows that the square $K L M N$ has area $3^{2}$, that is, 9 .

We leave it to the reader to show that the triangles $P K S, Q L P, R M Q$ and $S N R$ are congruent. It follows, in particular, that $S K=P L$. Therefore $S K$ has length 2 .
Using Pythagoras' Theorem, applied to the right-angled triangle $P K S$, we have $P S^{2}=1^{2}+2^{2}=5$. Therefore the square $P Q R S$ has area 5 .
Therefore the area of the square $P Q R S$ as a fraction of the area of the square $K L M N$ is $\frac{5}{9}$.
It follows, similarly, that the area of the square $T U V W$ as a fraction of the area of the square $P Q R S$ is also $\frac{5}{9}$.
It follows that the area of the shaded square $T U V W$ is $\frac{5}{9} \times \frac{5}{9} \times$ the area of the square $K L M N$.
Since $\frac{5}{9} \times \frac{5}{9}=\frac{25}{81}$,it follows that the fraction of the area of $K L M N$ that is shaded is $\frac{25}{81}$.
FOR INVESTIGATION
14.1 Show that the triangles $P K S, Q L P, R M Q$ and $S N R$ are congruent.
14.2 Suppose that $P$ divides the side $K L$ in the ratio $1: 3$, and that $T$ divides $P Q$ in the ratio $1: 3$.
In this case, what fraction of the area of $K L M N$ is shaded?
14.3 Suppose that $P$ divides the side $K L$ in the ratio $p: q$, and that $T$ divides $P Q$ in the ratio $r: s$.
In this case, what fraction of the area of $K L M N$ is shaded?