Question

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.}

• A. 10.5
• B. 11.0
• C. 12.0
• D. 12.5

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.

Answer 1

The Correct Option is C

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 \( \boxed{12 { km}} \).