GATE XE 2025 | Question: 10

Question

​Three villages $\text{P, Q}$, and $\text{R}$ are located in such a way that the distance $\text{PQ}=13 \mathrm{~km}$, $\mathrm{QR}=14 \mathrm{~km}$, and $\mathrm{RP}=15 \mathrm{~km}$, as shown in the figure. A straight road joins $\text{Q}$ and $\text{R}$. It is proposed to connect $\text{P}$ to this road $\text{QR}$ by constructing another road. What is the minimum possible length (in $\mathrm{~km}$) of this connecting road?

Note: The figure shown is representative.

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

Answer 1

According to Heron's formula

area (triangle) = sqrt{s(s-a)(s-b)(s-c)}

s=(a+b+c)/2

a=13, b=14, c=15, therefore s=21,  and area= 84

area of triangle = 1/2 (base*altitude)

84 = 1/2 (14*altitude)

therefore altitude = 12 (C)