Consider a triangular plot ABC with sides $\mathrm{AB}=7 \mathrm{~m}, \mathrm{BC}=5 \mathrm{~m}$ and $\mathrm{CA}=6 \mathrm{~m}$. A vertical lamp-post at the mid point D of AC subtends an angle $30^{\circ}$ at B. The height (in m) of the lamp-post is:

JEE Main · 2019 · 10 Jan 2019 · Shift 1 · Properties of Triangle

Consider a triangular plot ABC with sides

AB = 7 m, BC = 5 m

and

CA = 6 m

. A vertical lamp-post at the mid point D of AC subtends an angle

3 0^{\circ}

at B. The height (in m) of the lamp-post is:

Answer: (B)

The lamp-post stands at the midpoint D of AC, so its foot is at the far end of the median BD, whose length comes from the median formula

B D = \frac{1}{2} 2 (B C^{2} + B A^{2}) - A C^{2} = \frac{1}{2} 2 (25 + 49) - 36 = \frac{1}{2} 112 = 27

. The post subtends

3 0^{\circ}

at B, so

tan 3 0^{\circ} = \frac{h}{B D}

gives

h = 27 tan 3 0^{\circ} = \frac{2 7}{3} = \frac{2 21}{3}

JEE Main · 2019 · 10 Jan 2019 · Shift 1 · Single correct option