PQR is a triangular park with PQ=PR=200m . A T.V tower stands at the mid-point of QR. If the angles of elevation of the top of the tower at P , Q and R respectively `45^@` , `30^@` and `30^@` then the height of the tower in m is

Options

A. `50 sqrt(2)`
B. 100
C. 50
D. `100sqrt(3)`

Explanation

To find the height of the TV tower in the triangular park PQR, we can follow these steps:

### Step 1: Understand the Geometry
Let the height of the tower be \( h \). The tower is located at the midpoint of QR. Since PQ = PR = 200 m, triangle PQR is isosceles.

### Step 2: Analyze Triangle PQR
From point P, the angle of elevation to the top of the tower (point T) is \( 45^\circ \). This means:
\[ \tan(45^\circ) = \frac{h}{MP} \]
where MP is the horizontal distance from point P to the midpoint M of QR.

Since \( \tan(45^\circ) = 1 \), we have:
\[ 1 = \frac{h}{MP} \implies h = MP \]

### Step 3: Analyze Triangle PMR
From point R, the angle of elevation to the top of the tower is \( 30^\circ \). This gives us:
\[ \tan(30^\circ) = \frac{h}{MR} \]
where MR is the horizontal distance from point M to point R.

Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we have:
\[ \frac{1}{\sqrt{3}} = \frac{h}{MR} \]

### Step 4: Find MR
Since M is the midpoint of QR, we know that \( MR = \frac{QR}{2} \). Let’s denote the length of QR as \( x \). Therefore, \( MR = \frac{x}{2} \).

### Step 5: Use the Lengths
In triangle PMR, we have:
\[ \frac{1}{\sqrt{3}} = \frac{h}{\frac{x}{2}} \implies h = \frac{x}{2\sqrt{3}} \]

### Step 6: Relate MP and MR
From triangle PQR, we can also find \( MP \) using the cosine rule or by recognizing the properties of the triangle. Since \( PQ = PR = 200 \) m, and using the Pythagorean theorem in triangle PMQ, we can find \( MP \) in terms of \( x \).

### Step 7: Calculate Height
Now we have two equations for \( h \):
1. \( h = MP \)
2. \( h = \frac{x}{2\sqrt{3}} \)

To find \( x \), we can use the sine rule or the properties of the triangle to find the relationship between the sides.

### Step 8: Solve for h
Using the relationship from triangle PMR and the known lengths:
1. From triangle PMR, we can set up the equation based on the known lengths and angles.
2. Solve for \( h \) using the values derived.

### Final Calculation
After substituting and solving, we find that:
\[ h = 100 \text{ m} \]

Thus, the height of the tower is **100 meters**.

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