Line through the points `(-2,6)` and `(4,8)` is perpendicular to the line through the points `(8,12)` and `(x,24)`. Find the value of 'x'.

Answer

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

### Step-by-Step Solution:

1. **Find the slope of the line through points \((-2, 6)\) and \( (4, 8)\)**:
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 6}{4 - (-2)} = \frac{2}{6} = \frac{1}{3} \]

2. **Find the slope of the line through points \( (8, 12) \) and \( (x, 24) \)**:
\[ m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{24 - 12}{x - 8} = \frac{12}{x - 8} \]

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

4. **Solve for \( x \)**:
Multiply 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 by 3:
\[ x = 4 \]

### Final Answer:
The value of \( x \) is \( 4 \).