Question

Select the equation that is perpendicular to the two points below:

(-2,6) (-6,2)

a
y=x+9

b
y=2x-3

c
y=-x+8

Answer 1

To find the equation that is perpendicular to the line passing through the points \((-2, 6)\) and \((-6, 2)\), we first need to determine the slope of the line that connects these two points.

1. **Calculate the slope (m) of the line between the two points** using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, \((x_1, y_1) = (-2, 6)\) and \((x_2, y_2) = (-6, 2)\):
\[
m = \frac{2 - 6}{-6 + 2} = \frac{-4}{-4} = 1
\]

2. **Find the slope of the perpendicular line**. The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, if the slope \(m = 1\), the slope of the perpendicular line \(m_{\perpendicular}\) is:
\[
m_{\perpendicular} = -\frac{1}{1} = -1
\]

3. **Now, we can check the equations provided for a slope of -1**:

- Option a: \(y = x + 9\) has slope \(1\).
- Option b: \(y = 2x - 3\) has slope \(2\).
- Option c: \(y = -x + 8\) has slope \(-1\).

Only **option c**, \(y = -x + 8\), has a slope of \(-1\), which is perpendicular to the line joining the points \((-2, 6)\) and \((-6, 2)\).

Thus, the answer is:
**c: \(y = -x + 8\)**.