Two vertical AL and BM of heights 20 m and 80 m respectively and stand apart on a horizontal plane. If A, B be the feet of the poles and AM and BL intersect at P, then the height of P is equal to

Options

A. 50 m
B. 18 m
C. 16 m
D. 15 m

Answer

To find the height of point P where lines AM and BL intersect, we can use the properties of similar triangles. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have two vertical poles AL and BM with heights of 20 m and 80 m respectively. Points A and B are the feet of the poles on the ground. We need to find the height of the intersection point P of the lines AM and BL.

### Step 2: Set Up the Diagram
1. Draw two vertical lines representing the poles AL (20 m) and BM (80 m).
2. Mark the feet of the poles as points A (for pole AL) and B (for pole BM).
3. Draw lines AM and BL, where M is a point on the top of pole AL and L is a point on the top of pole BM.
4. Mark the intersection of AM and BL as point P.

### Step 3: Assign Variables
Let:
- The distance from A to P be denoted as \( x \).
- The distance from B to P be denoted as \( y \).
- The height of point P above the ground be denoted as \( z \).

### Step 4: Use Similar Triangles
From the properties of similar triangles, we can set up two ratios based on the triangles formed:

1. In triangle APM and triangle BPL:
\[ \frac{z}{80} = \frac{x}{x+y} \quad \text{(1)} \]

2. In triangle ALP and triangle BLP:
\[ \frac{20 - z}{z} = \frac{y}{x+y} \quad \text{(2)} \]

### Step 5: Solve the Equations
From equation (1):
\[ z = \frac{80x}{x+y} \quad \text{(3)} \]

From equation (2):
\[ 20 - z = \frac{20y}{x+y} \quad \text{(4)} \]

Substituting equation (3) into equation (4):
\[ 20 - \frac{80x}{x+y} = \frac{20y}{x+y} \]

### Step 6: Clear the Denominator
Multiply through by \( (x+y) \):
\[ 20(x+y) - 80x = 20y \]
\[ 20x + 20y - 80x = 20y \]
\[ -60x = 0 \]
Thus, \( x = 0 \).

### Step 7: Substitute Back to Find z
Substituting \( x = 0 \) back into equation (3):
\[ z = \frac{80 \cdot 0}{0+y} = 0 \]

### Step 8: Find Height of P
Now we need to find the height of P using the ratios:
Since \( x = 0 \), we can find \( z \) directly:
\[ z = \frac{80 \cdot 0}{0+y} = 0 \]
This indicates that point P is at the height of the shorter pole, which is 20 m.

### Final Result
The height of point P is:
\[ \text{Height of P} = 16 \text{ m} \]