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.
Show Hint
To find the shortest distance from a point to a line segment, drop a perpendicular and apply the Pythagorean theorem to form solvable right triangles.
Let the foot of the perpendicular from point P to line QR be at a distance \( x \) km from Q, and the perpendicular height be \( h \). We can now apply the Pythagorean theorem to two right-angled triangles:
\[
h^2 + x^2 = 13^2 = 169 \quad {(i)}
\]
\[
h^2 + (14 - x)^2 = 15^2 = 225 \quad {(ii)}
\]
Now subtract equation (i) from (ii):
\[
[h^2 + (14 - x)^2] - [h^2 + x^2] = 225 - 169
\]
\[
(14 - x)^2 - x^2 = 56
\]
\[
196 - 28x = 56 \Rightarrow 28x = 140 \Rightarrow x = 5
\]
Substitute \( x = 5 \) in equation (i):
\[
h^2 + 25 = 169 \Rightarrow h^2 = 144 \Rightarrow h = \sqrt{144} = 12
\]
Therefore, the minimum possible length of the connecting road is \( \boxed{12 { km}} \).
