Questions: Question 5 Line segment RH has endpoints R(-4,4) and H(2,-4). Which equation represents a line perpendicular to RH that passes through the point (3,-1) ?

Solution Steps

To find the slope of the line segment \( \overline{RH} \) with endpoints \( R(-4, 4) \) and \( H(2, -4) \), we use the slope formula:

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

Substituting the given points:

\[ m = \frac{-4 - 4}{2 - (-4)} = \frac{-8}{6} = -\frac{4}{3} \]

The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope of the line perpendicular to \( \overline{RH} \) is:

\[ m_{\perp} = -\frac{1}{-\frac{4}{3}} = \frac{3}{4} \]

We use the point-slope form of the equation of a line, which is:

\[ y - y_1 = m(x - x_1) \]

Given the point \( (3, -1) \) and the slope \( \frac{3}{4} \):

\[ y - (-1) = \frac{3}{4}(x - 3) \]

Simplifying, we get:

\[ y + 1 = \frac{3}{4}(x - 3) \]

Final Answer