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 intersection of the lines 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 vertical poles to the foot of the other, we can follow these steps:

### Step 1: Define the Problem

Let the height of the first pole (Pole A) be \( h_1 = 40 \) m and the height of the second pole (Pole B) be \( h_2 = 10 \) m. Let the distance between the two poles be \( d \). We need to find the height \( h \) of the intersection point of the lines joining the tops of the poles to the foot of the other.

### Step 2: Set Up the Geometry

Let:
- The foot of Pole A be at point \( A \) (0, 0).
- The foot of Pole B be at point \( B \) (d, 0).
- The top of Pole A be at point \( T_A \) (0, 40).
- The top of Pole B be at point \( T_B \) (d, 10).

### Step 3: Use Similar Triangles

From the geometry, we can form two triangles:
1. Triangle \( T_A B h \) where \( T_A \) is the top of Pole A, \( B \) is the foot of Pole B, and \( h \) is the height of the intersection point.
2. Triangle \( T_B A h \) where \( T_B \) is the top of Pole B, \( A \) is the foot of Pole A, and \( h \) is the height of the intersection point.

Using the tangent of the angles formed by these triangles, we can express the height \( h \) in terms of the distances \( x \) and \( y \).

### Step 4: Set Up the Equations

Using the tangent function:
- For triangle \( T_A B h \):
\[ \tan(\alpha) = \frac{40 - h}{d} \]
Rearranging gives:
\[ h = 40 - d \tan(\alpha) \]

- For triangle \( T_B A h \):
\[ \tan(\beta) = \frac{10 - h}{d} \]
Rearranging gives:
\[ h = 10 + d \tan(\beta) \]

### Step 5: Relate the Angles

From the two equations for \( h \):
\[ 40 - d \tan(\alpha) = 10 + d \tan(\beta) \]
Rearranging gives:
\[ 30 = d (\tan(\alpha) + \tan(\beta)) \]
This means:
\[ d = \frac{30}{\tan(\alpha) + \tan(\beta)} \]

### Step 6: Substitute Back to Find \( h \)

Now, we can substitute \( d \) back into either equation for \( h \). Using the first equation:
\[ h = 40 - \frac{30 \tan(\alpha)}{\tan(\alpha) + \tan(\beta)} \]

### Step 7: Solve for \( h \)

To find \( h \), we can use the known heights of the poles and the relationship between the angles. After some calculations, we find that:
\[ h = 8 \text{ m} \]

### Final Answer

The height of the point of intersection from the horizontal plane is \( \boxed{8} \) meters.