Question

In a triangular plot \(ABC\), the side lengths are \(BC=7m\), \(CA=8m\), and \(AB=9m\). A vertical lamp post is located at the midpoint \(E\) of side \(AC\). If the angle of elevation of the top of the lamp post from point \(B\) is \(\tan ^{-1}(4)\), then the height of the lamp post is...

• A. \(22\)
• B. \(24\)
• C. \(26\)
• D. \(28\)

Hint:

BE is the median to AC, so find BE using the median length formula and then use the angle of elevation.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
The lamp post stands at midpoint E of AC, so the horizontal distance from B is the median BE.

Step 2: Find the Median
Median formula: \(BE^{2}=\dfrac{2AB^{2}+2BC^{2}-AC^{2}}{4}\).
\(BE^{2}=\dfrac{2(81)+2(49)-64}{4}=\dfrac{196}{4}=49\), so \(BE=7\).

Step 3: Use the Elevation
If the height is h, then \(\tan\theta=\dfrac{h}{BE}\) with \(\tan\theta=4\).

Step 4: Height
\(h=4\times 7=28\) m.

Final Answer:
The height of the lamp post is 28 m, option (D). \[ \boxed{\text{(D) }28} \]