Two vertical poles of height 10 m and 40 m stand apart on a horizontal plane. The height (in meters) of the point of intersection of the line joining the top of each pole to the foot of the other, from this horizontal plane is

Options

A. 8
B. 10
C. 6
D. 4

Answer

To solve the problem of finding the height of the point of intersection of the lines joining the tops of two poles to the foot of the other, we can follow these steps:

### Step-by-Step Solution:

1. **Draw the Diagram**:
- Let the two poles be represented as points A (10 m) and C (40 m) on a horizontal plane.
- The foot of the first pole (A) is point B, and the foot of the second pole (C) is point D.
- The lines we are interested in are from the top of pole A to the foot of pole C (line AC) and from the top of pole C to the foot of pole A (line BD).

2. **Label the Points**:
- Let point E be the point of intersection of lines AC and BD.
- Let the height of point E from the horizontal plane be denoted as h.

3. **Use Similar Triangles**:
- In triangle ABE (formed by the top of pole A, foot B, and point E), we can apply the concept of similar triangles.
- The height of pole A (10 m) corresponds to the height at point E (h) and the distance from A to B (the base of pole A) can be denoted as x.
- Similarly, in triangle CDE (formed by the top of pole C, foot D, and point E), the height of pole C (40 m) corresponds to the height at point E (h) and the distance from C to D (the base of pole C) can be denoted as y.

4. **Set Up the Proportions**:
- From triangle ABE:
\[ \frac{h}{10} = \frac{x}{x+y} \]
- From triangle CDE:
\[ \frac{h}{40} = \frac{y}{x+y} \]

5. **Cross-Multiply and Rearrange**:
- From the first equation:
\[ h(x+y) = 10x \quad \text{(1)} \]
- From the second equation:
\[ h(x+y) = 40y \quad \text{(2)} \]

6. **Equate the Two Expressions for h**:
- Set the right-hand sides of equations (1) and (2) equal to each other:
\[ 10x = 40y \]
- Rearranging gives:
\[ x = 4y \quad \text{(3)} \]

7. **Substitute Back into One of the Proportions**:
- Substitute equation (3) into either equation (1) or (2). Using equation (1):
\[ h(x+y) = 10x \]
- Substitute \(x = 4y\):
\[ h(4y + y) = 10(4y) \]
\[ h(5y) = 40y \]
- Dividing both sides by y (assuming y ≠ 0):
\[ 5h = 40 \]
\[ h = 8 \]

8. **Conclusion**:
- The height of the point of intersection E from the horizontal plane is **8 meters**.