From the top of a building of height h m, the angle of depression of an object on the ground is `theta.` What is the distance (in m) of the object from the foot of the building?

Options

A. `h cot theta `
B. ` h tan theta `
C. ` h cos theta `
D. ` h sin theta `

Answer

To solve the problem, we need to find the distance of the object from the foot of the building given the height of the building (h meters) and the angle of depression (θ).

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have a building of height \( h \) meters.
- The angle of depression from the top of the building to the object on the ground is \( \theta \).
- The angle of depression is equal to the angle of elevation from the object to the top of the building, which is also \( \theta \).

2. **Identify the Right Triangle**:
- From the top of the building to the object on the ground, we can form a right triangle.
- The height of the building (perpendicular) is \( h \).
- The distance from the foot of the building to the object (base) is \( x \).

3. **Use the Tangent Function**:
- In a right triangle, the tangent of an angle is defined as the ratio of the opposite side (perpendicular) to the adjacent side (base).
- Therefore, we can write:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{x} \]

4. **Rearrange the Equation**:
- To find \( x \), we can rearrange the equation:
\[ x = \frac{h}{\tan(\theta)} \]

5. **Express in Terms of Cotangent**:
- We know that \( \cot(\theta) = \frac{1}{\tan(\theta)} \), so we can rewrite the equation as:
\[ x = h \cdot \cot(\theta) \]

### Final Answer:
The distance of the object from the foot of the building is:
\[ x = h \cdot \cot(\theta) \text{ meters} \]