Question
Find the slope of a line perpendicular to the line that passes through the points \( (7,10) \) and \( (13,8) \). The slope of a perpendicular line is \( \square \). (Type an integer or a simplified fraction.)
Ask by Edwards Peterson. in the United States
Mar 13,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The slope of a perpendicular line is \(3\).
Solution
1. Compute the slope of the line passing through the points \( (7,10) \) and \( (13,8) \) using the formula
\[
m=\frac{y_2-y_1}{x_2-x_1}.
\]
2. Substitute the given points into the formula:
\[
m=\frac{8-10}{13-7}=\frac{-2}{6}=-\frac{1}{3}.
\]
3. The slope of a line perpendicular to this one is the negative reciprocal of the slope. Thus, compute:
\[
m_{\perp}=-\frac{1}{m}=-\frac{1}{-\frac{1}{3}}=3.
\]
The slope of a perpendicular line is \(3\).
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Simplify this solution The Deep Dive
To find the slope of a line passing through the points \( (7,10) \) and \( (13,8) \), we can use the formula for the slope: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting in our points, we have: \[ m = \frac{8 - 10}{13 - 7} = \frac{-2}{6} = -\frac{1}{3} \] The slope of a line perpendicular to this line is the negative reciprocal of \(-\frac{1}{3}\). Therefore, we take: \[ \text{slope of perpendicular line} = -\frac{1}{(-\frac{1}{3})} = 3 \] So, the slope of a perpendicular line is \( 3 \).
