From the top of a cliff 20 m high, the angle of elevation of the top of a tower is found to be equal to the angle of depression of the foot of the tower. The height of the tower is:

Options

A. `20` m
B. 40 m
C. 60 m
D. 80 m

Answer

To solve the problem step by step, let's break it down clearly:

### Step 1: Understand the Problem
We have a cliff that is 20 meters high. From the top of this cliff, we need to find the height of a tower such that the angle of elevation to the top of the tower is equal to the angle of depression to the foot of the tower.

### Step 2: Draw a Diagram
1. Draw a vertical line representing the cliff (AB) with a height of 20 meters.
2. At point A (top of the cliff), draw a horizontal line to the right to represent the ground level.
3. From point A, draw a line upwards to point C (top of the tower) and downwards to point D (foot of the tower).
4. Let the height of the tower be H meters, and the distance from point A to point D (foot of the tower) be X meters.

### Step 3: Identify Angles
Let θ be the angle of elevation from point A to point C (top of the tower) and the angle of depression from point A to point D (foot of the tower). According to the problem, these angles are equal.

### Step 4: Set Up the Trigonometric Relationships
1. In triangle ABC (where AB is the height of the cliff and AC is the line of sight to the top of the tower):
\[ \tan(\theta) = \frac{H}{X} \quad \text{(1)} \]

2. In triangle ABD (where AB is the height of the cliff and AD is the line of sight to the foot of the tower):
\[ \tan(\theta) = \frac{20}{X} \quad \text{(2)} \]

### Step 5: Equate the Two Equations
Since both equations equal \(\tan(\theta)\), we can set them equal to each other:
\[ \frac{H}{X} = \frac{20}{X} \]

### Step 6: Solve for H
Since X is present in both denominators, we can multiply both sides by X (assuming X ≠ 0):
\[ H = 20 \]

### Step 7: Find the Total Height of the Tower
The total height of the tower (from the ground to the top of the tower) is:
\[ \text{Total Height} = \text{Height of the cliff} + H = 20 + 20 = 40 \text{ meters} \]

### Final Answer
The height of the tower is **40 meters**.

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