13.271. The race track for the bicycle competition is a contour of a right-angled triangle with a difference in the lengths of the legs of 2 km. The hypotenuse runs along a dirt road, while both legs are on the highway. One of the participants covered the segment along the dirt road at a speed of $30 \mathrm{km} / \mathrm{h}$, and both segments along the highway in the same time at a speed of $42 \mathrm{km} / \mathrm{h}$. Determine the length of the track
## Solution.

Length of the route

$S=AB+BC+AC=a+a+2+\sqrt{a^{2}+(a+2)^{2}}$ (see Fig. 13.12).

By the condition $\frac{\sqrt{a^{2}+(a+2)^{2}}}{30}=\frac{a+a+2}{42}$, from which $a=8 \text{ km}$.

$S=8+8+2+\sqrt{8^{2}+10^{2}}=24$ km.

Answer: 24 km.