Question

Line A passes through points (4,7) and (9,9). Line B passes through points (3,10) and (1,15). Are lines A and B parallel or perpendicular?

Answer 1

To determine if lines A and B are parallel or perpendicular, we need to find the slopes of both lines.

**Step 1: Find the slope of Line A**

The formula for the slope (\(m\)) given two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

For Line A, the points are \((4, 7)\) and \((9, 9)\):

\[
m_A = \frac{9 - 7}{9 - 4} = \frac{2}{5}
\]

**Step 2: Find the slope of Line B**

Using the same formula for Line B, with points \((3, 10)\) and \((1, 15)\):

\[
m_B = \frac{15 - 10}{1 - 3} = \frac{5}{-2} = -\frac{5}{2}
\]

**Step 3: Compare the slopes**

- Lines are **parallel** if their slopes are equal: \(m_A = m_B\).
- Lines are **perpendicular** if the product of their slopes is \(-1\): \(m_A \cdot m_B = -1\).

Now let's check the conditions:

- Check if they are parallel:
\[
\frac{2}{5} \neq -\frac{5}{2}
\]
So, they are not parallel.

- Check if they are perpendicular:
\[
m_A \cdot m_B = \left(\frac{2}{5}\right) \cdot \left(-\frac{5}{2}\right) = -1
\]

Since the product of the slopes is \(-1\), the lines are perpendicular.

**Conclusion: Lines A and B are perpendicular.**