ABC is a triangular park in the form of an equilateral triangle. A pillar at A subtends an angle of `45^(@)`. If `theta` be the angle of elevation of the pillar at D, the middle point of BC, then `tan theta` is equal to

Options

A. `sqrt(3)/2`
B. `2/sqrt(3)`
C. `sqrt(3)`h m
D. `1/sqrt(3)`

Answer

To solve the problem step-by-step, we need to find the value of \( \tan \theta \) where \( \theta \) is the angle of elevation of a pillar at point D, the midpoint of side BC of an equilateral triangle ABC.

### Step 1: Understand the Geometry
We have an equilateral triangle ABC with a pillar at point A. The angle subtended by the pillar at point A is \( 45^\circ \). Point D is the midpoint of side BC.

### Step 2: Set Up the Triangle
Let the side length of the equilateral triangle ABC be \( a \). Therefore, all sides are equal, i.e., \( AB = BC = CA = a \).

### Step 3: Find the Height of the Triangle
The height \( h \) of the equilateral triangle can be calculated using the formula:
\[ h = \frac{\sqrt{3}}{2} a \]

### Step 4: Determine the Coordinates
Assuming the triangle is positioned in the coordinate plane:
- Point A can be at \( (0, h) \)
- Point B can be at \( (-\frac{a}{2}, 0) \)
- Point C can be at \( (\frac{a}{2}, 0) \)
- Point D, being the midpoint of BC, will be at \( (0, 0) \).

### Step 5: Calculate the Height of the Pillar
Since the angle at A subtended by the pillar is \( 45^\circ \), we can use the tangent function:
\[ \tan 45^\circ = 1 = \frac{h}{AC} \]
Here, \( AC = a \). Therefore, we have:
\[ h = a \]

### Step 6: Find the Distance AD
Now, we need to find the distance \( AD \). Using the Pythagorean theorem in triangle ADC:
\[ AD^2 = AC^2 - DC^2 \]
Where \( AC = a \) and \( DC = \frac{a}{2} \) (since D is the midpoint of BC):
\[ AD^2 = a^2 - \left(\frac{a}{2}\right)^2 = a^2 - \frac{a^2}{4} = \frac{4a^2}{4} - \frac{a^2}{4} = \frac{3a^2}{4} \]
Thus,
\[ AD = \sqrt{\frac{3a^2}{4}} = \frac{a\sqrt{3}}{2} \]

### Step 7: Find \( \tan \theta \)
Now we can find \( \tan \theta \):
\[ \tan \theta = \frac{h}{AD} = \frac{a}{\frac{a\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

### Final Answer
Thus, the value of \( \tan \theta \) is:
\[ \tan \theta = \frac{2}{\sqrt{3}} \]