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

Explanation

To find the height of the lamp-post at the midpoint D of AC in triangular plot ABC, we can follow these steps:

### Step 1: Identify the Given Information
We have a triangle ABC with:
- AB = 7 m
- BC = 5 m
- CA = 6 m
- D is the midpoint of AC, so AD = CD = 3 m (since AC = 6 m).

### Step 2: Calculate the Length of the Median BD
To find the length of the median BD, we can use the formula for the median from vertex B to side AC:
\[ \text{Median} = \frac{1}{2} \sqrt{2A^2 + 2C^2 - B^2} \]
Here, we denote:
- A = AC = 6 m
- B = AB = 7 m
- C = BC = 5 m

Substituting the values:
\[ \text{Median} \, BD = \frac{1}{2} \sqrt{2(6^2) + 2(5^2) - (7^2)} \]
Calculating:
\[ = \frac{1}{2} \sqrt{2(36) + 2(25) - 49} \]
\[ = \frac{1}{2} \sqrt{72 + 50 - 49} \]
\[ = \frac{1}{2} \sqrt{73} \]
\[ = \frac{\sqrt{73}}{2} \]

### Step 3: Set Up the Right Triangle CDB
In triangle CDB, we know:
- Angle ∠CDB = 30°
- BD = \(\frac{\sqrt{73}}{2}\)
- We need to find height \(H\) (which is CD).

Using the tangent function:
\[ \tan(30°) = \frac{H}{BD} \]
Substituting the known values:
\[ \frac{1}{\sqrt{3}} = \frac{H}{\frac{\sqrt{73}}{2}} \]

### Step 4: Solve for Height H
Rearranging the equation to find H:
\[ H = \frac{\sqrt{73}}{2} \cdot \frac{1}{\sqrt{3}} \]
\[ H = \frac{\sqrt{73}}{2\sqrt{3}} \]

### Step 5: Rationalize the Denominator
To rationalize the denominator:
\[ H = \frac{\sqrt{73} \cdot \sqrt{3}}{2 \cdot 3} \]
\[ H = \frac{\sqrt{219}}{6} \]

### Final Answer
Thus, the height of the lamp-post is:
\[ H = \frac{\sqrt{219}}{6} \text{ m} \]