19. $P Q R S$ is a square. Point $T$ lies on $P Q$ so that $P T: T Q=1: 2$. Point $U$ lies on $S R$ so that $S U: U R=1: 2$. The perimeter of $P T U S$ is $40 \mathrm{~cm}$.
What is the area of PTUS?
A $40 \mathrm{~cm}^{2}$
B $45 \mathrm{~cm}^{2}$
C $48 \mathrm{~cm}^{2}$
D $60 \mathrm{~cm}^{2}$
E $75 \mathrm{~cm}^{2}$
19. E Let square $P Q R S$ have side $3 x \mathrm{~cm}$. Then, as $P T: T Q=1: 2$, $P T=x \mathrm{~cm}$. Similarly, $U S=x \mathrm{~cm}$. In triangles $P T S$ and $U S T, P T=U S, \angle P T S=\angle U S T$ (alternate angles) and $T S$ is common to both. So the triangles are congruent (SAS).
Therefore $U T=P S=3 x \mathrm{~cm}$ and $\angle T U S=\angle S P T=90^{\circ}$.
Hence $P T U S$ is a rectangle, which has perimeter $40 \mathrm{~cm}$. So
$40=2(3 x+x)=8 x$. Therefore $x=5$ and the area of $P T U S$, in $\mathrm{cm}^{2}$, is $15 \times 5=75$