18. In the diagram, $P Q R S$ is a square of side $10 \mathrm{~cm} . T$ is a point inside the square so that $\angle S P T=75^{\circ}$ and $\angle T S P=30^{\circ}$. What is the length of $T R$ ?
A $8 \mathrm{~cm}$
B $8.5 \mathrm{~cm}$
C $9 \mathrm{~cm}$
D $9.5 \mathrm{~cm}$
E $10 \mathrm{~cm}$
18. E Draw in lines $P T$ and $T S$ as shown. Since angles in a triangle add to $180^{\circ}$ and we are given $\angle S P T=75^{\circ}$ and $\angle T S P=30^{\circ}$, we obtain $\angle P T S=75^{\circ}$. Therefore $\triangle P T S$ is isosceles and hence $T S=P S=10 \mathrm{~cm}$. Therefore, since $R S=10 \mathrm{~cm}$ as it is a side of the square, $\triangle R S T$ is also isosceles. Since $\angle R S P=90^{\circ}$ and $\angle T S P=30^{\circ}$, we have $\angle R S T=60^{\circ}$. Therefore $\triangle R S T$ is isosceles with one angle equal to $60^{\circ}$. Hence $\triangle R S T$ is equilateral and therefore the length of $T R$ is $10 \mathrm{~cm}$.