Question

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 $m+b$.

Note: The perpendicular bisector of the line segment $\overline{AB}$ is the line that passes through the midpoint of $\overline{AB}$ and is perpendicular to $\overline{AB}$.

Answer 1

3.5

Explanation:

The attached shows the perpendicular bisector to have a slope of m = -1/2 and a y-intercept of b = 4.

m+b = (-1/2) +4 = 3.5

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The difference of coordinates of the two points is ...

B-A = (-5, 4) -(-3, 8) = (-2, -4)

Then the rise/run ratio is (-4)/(-2) = 2. The slope of the perpendicular bisector will have a slope (m) that is the negative reciprocal of this:

m = -1/2

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The average of the coordinates of the two points is ...

(B+A)/2 = ((-5, 4) +(-3, 8))/2 = (-8, 12)/2 = (-4, 6)

Then the point-slope form of the equation of the perpendicular bisector can be written ...

y = (-1/2)(x +4) +6 = (-1/2)x -2 +6

y = (-1/2)x +4

So, the y-intercept is b = 4, and ...

m + b = (-1/2) +4 = 3 1/2

m + b = 3.5