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 `30^(@)` at B. The height (in m) of the lamp-post is

Options

A. `2/3sqrt(21)`
B. `2sqrt(21)`
C. `7sqrt3`
D. `3/2sqrt(21)`

Explanation

To find the height of the lamp-post at the midpoint D of AC that subtends an angle of 30° at point B, we can follow these steps:

### Step 1: Identify the triangle and its sides
We have a triangle ABC with sides:
- AB = 7 m
- BC = 5 m
- CA = 6 m

### Step 2: Determine the midpoint D of AC
Since D is the midpoint of AC, we can find the lengths of AD and DC:
- AD = DC = CA / 2 = 6 m / 2 = 3 m

### Step 3: Calculate the length of median BD
To find the length of median BD, we can use the formula for the length of a median in a triangle:
\[ BD = \frac{1}{2} \sqrt{2AB^2 + 2BC^2 - AC^2} \]
Substituting the values:
- \(AB = 7\)
- \(BC = 5\)
- \(AC = 6\)

Calculating:
\[ BD = \frac{1}{2} \sqrt{2(7^2) + 2(5^2) - (6^2)} \]
\[ = \frac{1}{2} \sqrt{2(49) + 2(25) - 36} \]
\[ = \frac{1}{2} \sqrt{98 + 50 - 36} \]
\[ = \frac{1}{2} \sqrt{112} \]
\[ = \frac{1}{2} \cdot \sqrt{16 \cdot 7} = \frac{1}{2} \cdot 4\sqrt{7} = 2\sqrt{7} \]

### Step 4: Use the angle to find the height H of the lamp-post
We know that the angle ∠DBE = 30°. In triangle BDE, where E is the foot of the perpendicular from D to line BC, we can use the tangent function:
\[ \tan(30°) = \frac{H}{BD} \]
Since \(\tan(30°) = \frac{1}{\sqrt{3}}\), we have:
\[ \frac{1}{\sqrt{3}} = \frac{H}{BD} \]
Substituting \(BD = 2\sqrt{7}\):
\[ \frac{1}{\sqrt{3}} = \frac{H}{2\sqrt{7}} \]
Cross-multiplying gives:
\[ H = 2\sqrt{7} \cdot \frac{1}{\sqrt{3}} = \frac{2\sqrt{7}}{\sqrt{3}} \]

### Step 5: Rationalize the height H
To rationalize:
\[ H = \frac{2\sqrt{7}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{21}}{3} \]

### Final Answer
The height of the lamp-post is:
\[ H = \frac{2\sqrt{21}}{3} \text{ m} \]