21. In the diagram below, $A B C D$ is a convex quadrilateral such that $A C$ intersects $B D$ at the midpoint $E$ of $B D$. The point $H$ is the foot of the perpendicular from $A$ onto $D E$, and $H$ lies in the interior of the segment $D E$. Suppose $\angle B C A=90^{\circ}, C E=12 \mathrm{~cm}, E H=15 \mathrm{~cm}$, $A H=40 \mathrm{~cm}$ and $C D=x \mathrm{~cm}$. Find the value of $x$.
21. Answer: 40
Let $M$ be the midpoint of $A H$ and $N$ the midpoint of $B C$. Then $C D$ is parallel to $N E$ and $C D=2 N E$. Since $\triangle A H E$ is similar to $\triangle B C E$, we have $\triangle M H E$ is similar to $\triangle N C E$. As $M E=\sqrt{15^{2}+20^{2}}=25$, we thus have $N E=\frac{E O}{E H} \cdot M E=\frac{12 \times 25}{15}=20$. Therefore $C D=2 N E=40$.
None