Question

Find the slopes of the lines that passes through the given points. Then determine if the two lines are parallel or perpendicular or neither.

The line passing through the points (8,5) and (7,3).
The line passing through the points (6,-8) and (4,-7).
The slope of line 1 is Response area and the slope of line 2 is Response area. These lines are Response area.

Answer 1

To find the slopes of the lines that pass through the given points, we can use the slope formula:

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

**1. Finding the slope of the line passing through the points (8, 5) and (7, 3):**

Let \((x_1, y_1) = (8, 5)\) and \((x_2, y_2) = (7, 3)\).

Using the slope formula:

\[
m_1 = \frac{3 - 5}{7 - 8} = \frac{-2}{-1} = 2
\]

So, the slope of line 1 is \(m_1 = 2\).

**2. Finding the slope of the line passing through the points (6, -8) and (4, -7):**

Let \((x_1, y_1) = (6, -8)\) and \((x_2, y_2) = (4, -7)\).

Using the slope formula:

\[
m_2 = \frac{-7 - (-8)}{4 - 6} = \frac{-7 + 8}{4 - 6} = \frac{1}{-2} = -\frac{1}{2}
\]

So, the slope of line 2 is \(m_2 = -\frac{1}{2}\).

**Comparison of the slopes:**

- **Are the lines parallel?**: Two lines are parallel if their slopes are equal. Here, \(m_1 \neq m_2\), so the lines are not parallel.
- **Are the lines perpendicular?**: Two lines are perpendicular if the product of their slopes is \(-1\).

Calculating the product of the slopes:

\[
m_1 \cdot m_2 = 2 \cdot \left(-\frac{1}{2}\right) = -1
\]

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

To summarize:

- The slope of line 1 is **2**.
- The slope of line 2 is **-\frac{1}{2}**.
- These lines are **perpendicular**.