If the line joining the points `(6,-2)` and `(8,4)` is perpendicular to the line joining the points `(12,8)` and `(24,y)`, find the value of `y`.

Answer

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

### Step 1: Calculate the slope of the line joining points \( (6, -2) \) and \( (8, 4) \).

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 \( (6, -2) \) and \( (8, 4) \):

- \( x_1 = 6, y_1 = -2 \)
- \( x_2 = 8, y_2 = 4 \)

Substituting these values into the slope formula:

\[ m_{AB} = \frac{4 - (-2)}{8 - 6} = \frac{4 + 2}{2} = \frac{6}{2} = 3 \]

### Step 2: Calculate the slope of the line joining points \( (12, 8) \) and \( (24, y) \).

Using the same slope formula for points \( (12, 8) \) and \( (24, y) \):

- \( x_1 = 12, y_1 = 8 \)
- \( x_2 = 24, y_2 = y \)

The slope \( m_{PQ} \) is:

\[ m_{PQ} = \frac{y - 8}{24 - 12} = \frac{y - 8}{12} \]

### Step 3: Set up the equation for perpendicular lines.

For two lines to be perpendicular, the product of their slopes must equal \(-1\):

\[ m_{AB} \cdot m_{PQ} = -1 \]

Substituting the slopes we found:

\[ 3 \cdot \frac{y - 8}{12} = -1 \]

### Step 4: Solve for \( y \).

Multiply both sides by 12 to eliminate the denominator:

\[ 3(y - 8) = -12 \]

Now, divide both sides by 3:

\[ y - 8 = -4 \]

Adding 8 to both sides gives:

\[ y = 4 \]

### Conclusion

The value of \( y \) is \( 4 \).

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