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