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

Explanation

To determine whether 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 calculate the slopes of both lines and check if the product of the slopes equals -1.

### Step-by-Step Solution:

**Step 1: Find the slope of the first line (m1)**
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 the points (3, -4) and (-2, 6):
- Let \((x_1, y_1) = (3, -4)\) and \((x_2, y_2) = (-2, 6)\).
- Substitute the values into the slope formula:
\[ m_1 = \frac{6 - (-4)}{-2 - 3} = \frac{6 + 4}{-5} = \frac{10}{-5} = -2 \]

**Step 2: Find the slope of the second line (m2)**
Using the same slope formula for the points (-3, 6) and (9, -18):
- Let \((x_1, y_1) = (-3, 6)\) and \((x_2, y_2) = (9, -18)\).
- Substitute the values into the slope formula:
\[ m_2 = \frac{-18 - 6}{9 - (-3)} = \frac{-24}{9 + 3} = \frac{-24}{12} = -2 \]

**Step 3: Check if the lines are perpendicular**
For two lines to be perpendicular, the product of their slopes must equal -1:
\[ m_1 \cdot m_2 = -2 \cdot -2 = 4 \]

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

### Conclusion:
The statement that the line joining the points (3, -4) and (-2, 6) is perpendicular to the line joining the points (-3, 6) and (9, -18) is **false**.

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