The line joining the points (2, x) and (3, 1) is perpendicular to the line joining the points (x, 4) and (7, 5). The value of x is

Options

A. 2
B. 3
C. 4
D. 7

Answer

To solve the problem, we need to find the value of \( x \) such that the line joining the points \( (2, x) \) and \( (3, 1) \) is perpendicular to the line joining the points \( (x, 4) \) and \( (7, 5) \).

### Step-by-step Solution:

1. **Identify the points**:
- Let point A be \( (2, x) \) and point B be \( (3, 1) \).
- Let point C be \( (x, 4) \) and point D be \( (7, 5) \).

2. **Calculate the slope of line AB**:
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 A and B:
\[ m_{AB} = \frac{1 - x}{3 - 2} = 1 - x \]

3. **Calculate the slope of line CD**:
For points C and D:
\[ m_{CD} = \frac{5 - 4}{7 - x} = \frac{1}{7 - x} \]

4. **Set up the perpendicularity condition**:
Since the lines are perpendicular, the product of their slopes should equal \(-1\):
\[ m_{AB} \cdot m_{CD} = -1 \]
Substituting the slopes we found:
\[ (1 - x) \cdot \frac{1}{7 - x} = -1 \]

5. **Cross-multiply to eliminate the fraction**:
\[ 1 - x = - (7 - x) \]
Simplifying this gives:
\[ 1 - x = -7 + x \]

6. **Rearranging the equation**:
Bringing all terms involving \( x \) to one side and constants to the other side:
\[ 1 + 7 = x + x \]
\[ 8 = 2x \]

7. **Solve for \( x \)**:
\[ x = \frac{8}{2} = 4 \]

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

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