15. Given a rectangle $A B C D, X$ and $Y$ are respectively points on $A B$ and $B C$. Suppose the areas of the triangles $\triangle A X D$, $\triangle B X Y$ and $\triangle D Y C$ are respectively 5,4 and 3 . Find the area of $\triangle D X Y$.
(2 marks)
15. $A B C D$ 為長方形, $X$ 和 $Y$ 分別為 $A B$ 和 $B C$ 上的點。若 $\triangle A X D 、 \triangle B X Y$ 和 $\triangle D Y C$ 的面積分別為 $5 、 4 、 3$,求 $\triangle D X Y$ 的面積。
15. $2 \sqrt{21}$
15. Let $A X=a$ and $X B=b$. Then $A D=\frac{10}{a}$ and $B Y=\frac{8}{b}$. Now the area of the rectangle is $\frac{10}{a}(a+b)=10+10 R$, where $R=\frac{b}{a}$ is to be determined. The area of $\triangle D X Y$ is then equal to $10+10 R-5-4-3=10 R-2$.
Consider $\triangle D Y C$. We have
$$
6=D C \times C Y=(a+b)\left(\frac{10}{a}-\frac{8}{b}\right)=2+10 R-\frac{8}{R} .
$$

It follows that $5 R^{2}-2 R-4=0$. This gives $R=\frac{2 \pm \sqrt{2^{2}-4(5)(-4)}}{2(5)}$. We take the positive root since $R$ is positive. The answer is $10 R-2=2 \sqrt{21}$.