7. The diagram shows a triangle $A B C$ with area $12 \mathrm{~cm}^{2}$. The sides of the triangle are extended to points $P, Q, R, S, T$ and $U$ as shown so that $P A=A B=B S, Q A=A C=C T$ and $R B=B C=C U$.

What is the area (in $\mathrm{cm}^{2}$ ) of hexagon $P Q R S T U$ ?
7. 156 Consider triangles $A B C$ and $A S T$. Angles $C A B$ and $T A S$ are equal because they are the same angle, $S A=2 B A$ and $T A=2 C A$. Hence triangles $A B C$ and AST are similar. The ratio of their sides is $1: 2$ and hence the ratio of their areas is $1^{2}: 2^{2}=1: 4$. Therefore the area of triangle AST is $4 \times 12 \mathrm{~cm}^{2}=48 \mathrm{~cm}^{2}$ and hence the area of BSTC is $(48-12) \mathrm{cm}^{2}=36 \mathrm{~cm}^{2}$. In a similar way, it can be shown that each of the areas of $C U P A$ and
$A Q R B$ is also $36 \mathrm{~cm}^{2}$. Next consider triangles $A B C$ and $A P Q$. Angles $B A C$ and $P A Q$ are equal using vertically opposite angles, $A B=A P$ and $A C=A Q$. Hence triangles $A B C$ and $A P Q$ are congruent (SAS) and so the area of triangle $A P Q$ is $12 \mathrm{~cm}^{2}$. In a similar way, it can be shown that the each of areas of triangles BRS and $C T U$ is also $12 \mathrm{~cm}^{2}$. Hence the total area of hexagon $P Q R S T U$ in $\mathrm{cm}^{2}$ is $(3 \times 36+4 \times 12)=156$.