17. A hare and a tortoise competed in a $5 \mathrm{~km}$ race along a straight line, going due North. The hare is five times as fast as the tortoise. The hare mistakenly started running due East. After a while he realised his mistake, then turned and ran straight to the finish point. He arrived at the same time as the tortoise. What was the distance between the hare's turning point and the finish point?
A $11 \mathrm{~km}$
B $12 \mathrm{~km}$
C $13 \mathrm{~km}$
D $14 \mathrm{~km}$
E $15 \mathrm{~km}$
Solution
$\mathbf{C}$
The routes of the hare and the tortoise form a right-angled triangle as shown on the diagram. The hare travels 5 times as fast, but arrives at the same time as the tortoise, so has travelled five times further, giving
$$
x+y=25 \text {. [1] }
$$

Pythagoras' Theorem gives $x^{2}+5^{2}=y^{2}$, so $y^{2}-x^{2}=25$, and this factorises to give $(y+x)(y-x)=25$. However, $x+y=25$ by equation [1]. So $25(y-x)=25$ and then $y-x=1$, and $y=x+1$. Substituting this into equation [1] gives $x+x+1=25$, and hence $x=12$ and $y=13$.