Question

Line r passes through points (10,9) and (3,14). line s is perpendicular to r. what is the slope of line s? simplify and write as a proper fraction, improper fraction, or integer.

Answer 1

To find the slope of line \( s \) which is perpendicular to line \( r \), we first need to determine the slope of line \( r \).

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 points \( (10, 9) \) and \( (3, 14) \):

- \( (x_1, y_1) = (10, 9) \)
- \( (x_2, y_2) = (3, 14) \)

Substituting these values into the slope formula:

\[
m_r = \frac{14 - 9}{3 - 10} = \frac{5}{-7} = -\frac{5}{7}
\]

Now we know that the slope of line \( r \) is \( -\frac{5}{7} \).

The slope of a line \( s \) that is perpendicular to line \( r \) is the negative reciprocal of the slope of line \( r \). The negative reciprocal of \( m_r \) is calculated as follows:

1. Take the reciprocal:
\[
\text{Reciprocal of } -\frac{5}{7} = -\frac{7}{5}
\]

2. Change the sign (to get the negative reciprocal):
\[
\text{Negative reciprocal} = \frac{7}{5}
\]

Therefore, the slope of line \( s \) is:

\[
\boxed{\frac{7}{5}}
\]