An adult and a minor boy, standing on the ground, are 4 metres apart. The height of the adult is 4 times the height of the minor boy. If at the mid-point of the line segment joining their feet, the angles of elevation of their tops are complementary, then the height of the minor boy is

Options

A. 1.3 m
B. 1.5 m
C. 1 m
D. 1.2 m

Answer

To solve the problem step-by-step, we will denote the height of the minor boy as \( h \). Consequently, the height of the adult will be \( 4h \).

### Step 1: Understand the setup
The adult and the minor boy are standing 4 meters apart. The midpoint between them is 2 meters from each of them.

### Step 2: Define the angles of elevation
Let \( x \) be the angle of elevation from the midpoint to the top of the adult, and \( y \) be the angle of elevation to the top of the minor boy. Since the angles are complementary, we have:
\[ x + y = 90^\circ \]

### Step 3: Set up the triangles
From the midpoint (M), we can form two right triangles:
1. Triangle ABM (where A is the top of the adult, B is the foot of the adult, and M is the midpoint)
2. Triangle CDM (where C is the top of the minor boy, D is the foot of the minor boy, and M is the midpoint)

### Step 4: Write the tangent equations
For triangle ABM:
\[ \tan(x) = \frac{\text{height of adult}}{\text{base}} = \frac{4h}{2} = 2h \]
Thus, we have:
\[ \tan(x) = 2h \quad \text{(Equation 1)} \]

For triangle CDM:
\[ \tan(y) = \frac{\text{height of minor boy}}{\text{base}} = \frac{h}{2} \]
Thus, we have:
\[ \tan(y) = \frac{h}{2} \quad \text{(Equation 2)} \]

### Step 5: Use the complementary angle identity
Since \( x + y = 90^\circ \), we can use the identity:
\[ \tan(x + y) = \tan(90^\circ) \rightarrow \text{undefined (infinity)} \]
This leads us to:
\[ 1 - \tan(x) \tan(y) = 0 \]
Substituting the values from Equation 1 and Equation 2:
\[ 1 - (2h) \left(\frac{h}{2}\right) = 0 \]

### Step 6: Solve for \( h \)
Simplifying the equation:
\[ 1 - h^2 = 0 \]
This implies:
\[ h^2 = 1 \]
Taking the square root:
\[ h = 1 \text{ meter} \]

### Conclusion
The height of the minor boy is \( 1 \) meter.

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