7. A triangle has two medians of lengths 9 and 12. Find the largest possible area of the triangle. (Note: A median is a line segment joining a vertex of the triangle to the midpoint of the opposite side.)
(1 mark)
某三角形其中兩條中線的長度為 9 和 12 。求該三角形的最大可能面積。（註：中線是指連起三角形一個頂點及其對邊的中點的線段。)
7. 72
7. As shown in the figure, let the medians $A D, B E$ and $C F$ of $\triangle A B C$ meet at the centroid $G$. Recall that the medians divide $\triangle A B C$ into 6 smaller triangles of equal area and that the centroid divides each median in the ratio $2: 1$.

Let $A D=9$ and $B E=12$. Then $A G=9 \times \frac{2}{3}=6$ and $B G=12 \times \frac{2}{3}=8$. Hence the area of $\triangle A B G$ is
$$
\frac{1}{2} \times A G \times B G \times \sin \angle A G B=\frac{1}{2} \times 6 \times 8 \times \sin \angle A G B=24 \sin \angle A G B \leq 24,
$$
where equality is possible when $\angle A G B=90^{\circ}$. Since the area of $\triangle A B G$ is $\frac{1}{3}$ the area of $\triangle A B C$, the largest possible area of $\triangle A B C$ is $24 \times 3=72$.