17. If a square can completely cover a triangle with side lengths 3,4 and 5 , find the smallest possible side length of the square.
(2 marks)
若一個正方形可以完全覆蓋一個邊長 3、4、5 的三角形, 求正方形的最小邊長。
(2 分)
17. $\frac{16 \sqrt{17}}{17}$
17. Suppose that in $\triangle A B C, B C=3, A C=4$ and $A B=5$. Of course $\angle A C B=90^{\circ}$. Without loss of generality, we fix $A$ at the origin and assume that $\triangle A B C$ lies in the first quadrant, and that the square to enclose $\triangle A B C$ has sides parallel to the coordinate axes and has $A$ as a vertex. Let $A D E F$ denote the smallest rectangle with sides parallel to the coordinate axes that can enclose $\triangle A B C$. Clearly, when $A D E F$ is a square, that is the smallest square that can enclose $\triangle A B C$.

Now suppose $A D E F$ is indeed a square with $C$ on $D E$ and $B$ on $E F$ as shown. Let $\angle C A D=\angle B C E=\theta$. We have $A D=4 \cos \theta$ and $E D=3 \cos \theta+4 \sin \theta$. Equating, we get $\cos \theta=4 \sin \theta$ and hence $\tan \theta=\frac{1}{4}$. Thus $A D=4 \cos \theta=4 \times \frac{4}{\sqrt{17}}=\frac{16 \sqrt{17}}{17}$, which is the desired smallest possible side length.