Discussion
Three villages P, Q, and R are located in such a way that the distance PQ = 13 km, QR = 14 km, and RP = 15 km, as shown in the figure. A straight road joins Q and R. It is proposed to connect P to this road QR by constructing another road. What is the minimum possible length (in km) of this connecting road?
Note: The figure shown is representative.
Answer: C
The shortest road is 12.0 km, the perpendicular from P to QR.
The shortest path from a point to a line is the perpendicular, so we need the altitude from P onto QR. Find the triangle's area with Heron's formula, where \(s = (13 + 14 + 15)/2 = 21\):
\[ \begin{aligned} \text{Area} &= \sqrt{21 \times 8 \times 7 \times 6} \\ &= \sqrt{7056} = 84\ \text{km}^{2} \\ h &= \frac{2 \times 84}{14} = 12\ \text{km} \end{aligned} \]
The foot of the perpendicular falls between Q and R, 5 km from Q (since \(\sqrt{13^{2} - 12^{2}} = 5\)), so the road meets QR itself. No road can be shorter than this perpendicular, which rules out 10.5 and 11.0 km, and 12.5 km is longer than the minimum.
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