Suppose angle of depression from top of the tower to point A is `45^(@)` and height of tower is 26 m. What is the distance of point A from the building?

Options

A. 25 m
B. 26 m
C. 31 m
D. 18 m

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have a tower of height 26 meters, and we need to find the distance from the base of the tower to point A, given that the angle of depression from the top of the tower to point A is 45 degrees.

### Step 2: Draw the Diagram
Draw a vertical line representing the tower (let's call it BC), where B is the top of the tower and C is the base. Point A is on the ground at some distance from the base of the tower. The angle of depression from point B to point A is 45 degrees.

### Step 3: Identify Angles
Since the angle of depression from B to A is 45 degrees, the angle of elevation from A to B is also 45 degrees (alternate interior angles).

### Step 4: Use Right Triangle Properties
In triangle ABC:
- AB is the height of the tower (26 m).
- AC is the distance from the base of the tower to point A (which we need to find).
- Angle ABC is 90 degrees (as it is a right triangle).
- Angle ACB is 45 degrees.

### Step 5: Apply Trigonometric Ratios
We can use the tangent function, which is defined as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Here, \(\theta\) is 45 degrees, the opposite side is AB (height of the tower = 26 m), and the adjacent side is AC (distance from the building to point A).

So we have:
\[ \tan(45^\circ) = \frac{AB}{AC} \]
Substituting the known values:
\[ \tan(45^\circ) = \frac{26}{AC} \]

### Step 6: Solve for AC
We know that \(\tan(45^\circ) = 1\). Therefore:
\[ 1 = \frac{26}{AC} \]
Cross-multiplying gives:
\[ AC = 26 \text{ m} \]

### Conclusion
The distance of point A from the building is 26 meters.

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