If a line joining the points `(-2,6)` and (4,8) is perpendicular to the line joining the points (8,12) and (x,24), then the value of x is
(i) 3
(ii) 4
(iii) -4
(iv) 2

Options

A. 3
B. 4
C. `-4`
D. 2

Answer

To solve the problem, we need to find the value of \( x \) such that the line joining the points \((-2, 6)\) and \((4, 8)\) is perpendicular to the line joining the points \((8, 12)\) and \((x, 24)\).

### Step 1: Calculate the slope of the line joining points \((-2, 6)\) and \((4, 8)\).
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For points \((-2, 6)\) and \((4, 8)\):
- \(x_1 = -2\), \(y_1 = 6\)
- \(x_2 = 4\), \(y_2 = 8\)

Substituting these values into the slope formula:
\[ m_1 = \frac{8 - 6}{4 - (-2)} = \frac{2}{4 + 2} = \frac{2}{6} = \frac{1}{3} \]

### Step 2: Calculate the slope of the line joining points \((8, 12)\) and \((x, 24)\).
Using the same slope formula for points \((8, 12)\) and \((x, 24)\):
- \(x_1 = 8\), \(y_1 = 12\)
- \(x_2 = x\), \(y_2 = 24\)

Substituting these values into the slope formula:
\[ m_2 = \frac{24 - 12}{x - 8} = \frac{12}{x - 8} \]

### Step 3: Set up the equation for perpendicular lines.
Since the lines are perpendicular, the product of their slopes must equal \(-1\):
\[ m_1 \cdot m_2 = -1 \]
Substituting the slopes we found:
\[ \frac{1}{3} \cdot \frac{12}{x - 8} = -1 \]

### Step 4: Solve for \( x \).
Multiplying both sides by \(3(x - 8)\) to eliminate the fraction:
\[ 12 = -3(x - 8) \]
Expanding the right side:
\[ 12 = -3x + 24 \]
Rearranging gives:
\[ 3x = 24 - 12 \]
\[ 3x = 12 \]
Dividing both sides by 3:
\[ x = 4 \]

### Conclusion
The value of \( x \) is \( 4 \).