9. Triangles $X Y Z$ and $P Q R$ are drawn on a square grid. What fraction of the area of triangle $X Y Z$ is the area of triangle $P Q R$ ?
A $\frac{1}{4}$
B $\frac{7}{18}$
C $\frac{1}{2}$
D $\frac{5}{18}$
E $\frac{1}{3}$
SOlUTION
E We assume that we have chosen units so that the sides of the squares in the grid have length 1 . We calculate the areas of the triangles $P Q R$ and $X Y Z$ by using the formula
$$
\text { area }=\frac{1}{2}(\text { base } \times \text { height })
$$
for the area of a triangle.
The triangle $P Q R$ has base $P R$ of length 3, and height $Q R$ of length 2. So the area of triangle $P Q R$ is $\frac{1}{2}(3 \times 2)=3$. The triangle $X Y Z$ has base $X Z$ of length 6 and height $Y R$ of length 3. So the area of triangle $X Y Z$ is $\frac{1}{2}(6 \times 3)=9$. Therefore the area of triangle $P Q R$ is $\frac{3}{9}=\frac{1}{3}$ of the area of triangle $X Y Z$.