A person standing on the bank of a river finds that the angle of elevation of the top of a tower on the opposite bank is `45^@`, then which of the following statements is correct ?

Options

A. Breadth of the river is twice the height of the tower
B. Breadth of the river and the height of the tower are the same
C. Breadth of the river is half of the height of the tower
D. none of these

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Problem
We have a person standing on one bank of a river, observing a tower on the opposite bank. The angle of elevation to the top of the tower is given as 45 degrees.

### Step 2: Define the Variables
Let:
- \( AB \) be the height of the tower (denoted as \( h \)).
- \( BC \) be the width of the river (denoted as \( x \)).
- The point \( C \) is where the person is standing.

### Step 3: Draw the Diagram
Draw a right triangle where:
- \( AB \) is the vertical side (height of the tower).
- \( BC \) is the horizontal side (width of the river).
- The angle \( \angle ACB \) is 45 degrees.

### Step 4: Apply Trigonometric Ratios
Using the tangent function for the angle of elevation:
\[ \tan(45^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} \]
Since \( \tan(45^\circ) = 1 \), we can write:
\[ 1 = \frac{h}{x} \]

### Step 5: Solve the Equation
From the equation \( 1 = \frac{h}{x} \), we can deduce:
\[ h = x \]
This means that the height of the tower is equal to the width of the river.

### Step 6: Conclusion
The correct statement is that the height of the tower is equal to the width of the river.