Line joining the points (3,-4) and (-2,6) is perpendicular to the line joining the points (-3,6) and (9,-18).

Answer

To determine if the line joining the points (3, -4) and (-2, 6) is perpendicular to the line joining the points (-3, 6) and (9, -18), we need to find the slopes of both lines and check if the product of their slopes is -1.

### Step 1: Find the slope of the line joining points (3, -4) and (-2, 6).

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 (3, -4) and (-2, 6):

- \( x_1 = 3 \)
- \( y_1 = -4 \)
- \( x_2 = -2 \)
- \( y_2 = 6 \)

Substituting these values into the slope formula:

\[ m_{PQ} = \frac{6 - (-4)}{-2 - 3} = \frac{6 + 4}{-5} = \frac{10}{-5} = -2 \]

### Step 2: Find the slope of the line joining points (-3, 6) and (9, -18).

Using the same slope formula for points (-3, 6) and (9, -18):

- \( x_1 = -3 \)
- \( y_1 = 6 \)
- \( x_2 = 9 \)
- \( y_2 = -18 \)

Substituting these values:

\[ m_{AB} = \frac{-18 - 6}{9 - (-3)} = \frac{-24}{9 + 3} = \frac{-24}{12} = -2 \]

### Step 3: Check if the lines are perpendicular.

For two lines to be perpendicular, the product of their slopes should be -1:

\[ m_{PQ} \cdot m_{AB} = (-2) \cdot (-2) = 4 \]

Since \( 4 \neq -1 \), the lines are not perpendicular.

### Conclusion:

The line joining the points (3, -4) and (-2, 6) is **not** perpendicular to the line joining the points (-3, 6) and (9, -18).

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