The perpendicular bisector of the line segment connecting the points $(-3,8)$ and $(-5,4)$ has an equation of the form $y = mx + b$. Find $m+b$.
the midpoint of the segment is ...(-4,6)
the slope of the segment is ... 2
... the slope of the perpendicular is the negative-reciprocal ... m = -1/2
using point-slope ... y - 6 = -1/2 (x - -4)
y = -1/2 x + 4
To find the equation of the perpendicular bisector of a line segment connecting two points, we need to follow these steps:
Step 1: Find the midpoint of the line segment.
Step 2: Find the slope of the line segment.
Step 3: Find the negative reciprocal of the slope.
Step 4: Use the midpoint and the negative reciprocal slope to find the equation of the perpendicular bisector.
Let's apply these steps to solve the problem.
Step 1: Find the midpoint of the line segment.
The coordinates of the two given points are $(-3,8)$ and $(-5,4)$. To find the midpoint, we can use the formula:
$$\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$
Using the coordinates, we find the midpoint as follows:
$$\left(\frac{-3 + (-5)}{2}, \frac{8 + 4}{2}\right) = (-4, 6)$$
Step 2: Find the slope of the line segment.
We can use the formula for slope, which is given by:
$$\text{Slope} = \frac{y_2-y_1}{x_2-x_1}$$
Using the coordinates of the given points $(-3,8)$ and $(-5,4)$, we find the slope as follows:
$$\text{Slope} = \frac{4-8}{-5-(-3)} = -\frac{4}{2} = -2$$
Step 3: Find the negative reciprocal of the slope.
The negative reciprocal of a number is the negative of its reciprocal. The reciprocal of $-2$ is $-\frac{1}{2}$, so the negative reciprocal is $-(-\frac{1}{2}) = \frac{1}{2}$.
Step 4: Use the midpoint and the negative reciprocal slope to find the equation of the perpendicular bisector.
Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, we can substitute the values of the midpoint $(-4, 6)$ and the negative reciprocal slope $\frac{1}{2}$ into the equation:
$$y - 6 = \frac{1}{2}(x - (-4))$$
Simplifying, we get:
$$y - 6 = \frac{1}{2}(x + 4)$$
$$2(y - 6) = x + 4$$
$$2y - 12 = x + 4$$
$$2y = x + 16$$
Dividing both sides by 2, we have:
$$y = \frac{x}{2} + 8$$
From the equation $y = \frac{x}{2} + 8$, we can see that $m = \frac{1}{2}$ and $b = 8$. Therefore, $m + b = \frac{1}{2} + 8 = \frac{17}{2}$. So the value of $m + b$ is $\frac{17}{2}$ or $8.5$.