24. The diagram shows triangle $J K L$ of area $S$. The point $M$ is the midpoint of $K L$. The points $P, Q, R$ lie on the extended lines $L J, M J, K J$, respectively, such that $J P=2 \times J L, J Q=3 \times J M$ and $J R=4 \times J K$.
What is the area of triangle $P Q R$ ?
A $S$
B $2 S$
C $3 S$
D $\frac{1}{2} S$
E $\frac{1}{3} S$
24. A Let $u$ be the area of triangle $J L M$, and $v$ be the area of triangle $J M K$. Since $M$ is the midpoint of $K L$, the triangle $J L M$ has half the area of the triangle $J L K$, so $u=v=\frac{1}{2} S$.
Note that $\angle Q J R=\angle K J M, J R=4 \times J K$ and $J Q=3 \times J M$. Hence, using the formula "Area of a triangle $=\frac{1}{2} a b \sin C$ ", we see that the area of triangle $J Q R=3 \times 4 \times v=6 S$.
Similarly the area of triangle $P J Q=2 \times 3 \times u=6 u=$ $3 S$ and the area of triangle $J P R=2 \times 4 \times S=8 S$. Hence the area of triangle $P Q R=6 S+3 S-8 S=S$.