At a point 15 metres away from the base of a 15 metres high house, the angle of elevation of the top is

Options

A. `45^(@)`
B. `30^(@)`
C. `60^(@)`
D. `90^(@)`

Answer

To solve the problem step by step, we will use trigonometric ratios to find the angle of elevation from a point 15 meters away from the base of a 15-meter high house.

### Step 1: Understand the scenario
We have a right triangle formed by:
- The height of the house (perpendicular) = 15 meters
- The distance from the point to the base of the house (base) = 15 meters

### Step 2: Identify the right triangle
Let:
- Point A be the point from where the angle of elevation is measured.
- Point B be the top of the house.
- Point C be the base of the house.

The triangle ABC is a right triangle where:
- AB = height of the house = 15 meters (perpendicular)
- AC = distance from point A to the base of the house = 15 meters (base)

### Step 3: Use the tangent ratio
The angle of elevation θ can be found using the tangent function:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{AC} \]
Substituting the values:
\[ \tan(\theta) = \frac{15}{15} \]

### Step 4: Simplify the equation
\[ \tan(\theta) = 1 \]

### Step 5: Find the angle θ
From trigonometric tables or knowledge:
\[ \tan(45^\circ) = 1 \]
Thus, we can conclude:
\[ \theta = 45^\circ \]

### Step 6: Conclusion
The angle of elevation of the top of the house from the point 15 meters away is \( 45^\circ \).