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GATE 2025 CE (CE2) – Question 10

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

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. 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.

  1. 10.5
  2. 11.0
  3. 12.0
  4. 12.5

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Show answer and explanation

Correct answer: (C) 12.0

Explanation

The shortest connection is the perpendicular from P to QR, that is the height of the triangle on the base QR. The sides are 13, 14 and 15, so $s = 21$ and the area is $\sqrt{21 \times 8 \times 7 \times 6} = 84$ km². The height is $\frac{2 \times 84}{14} = 12$ km (the foot lies inside QR, since the triangle has no obtuse angle).