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Difficulty: HardParallel and Perpendicular Lines

In the standard (x,y)(x, y) coordinate plane, the perpendicular bisector of the line segment with endpoints D(−5,8)D(-5, 8) and E(3,4)E(3, 4) is represented by the equation y=mx+by = mx + b. What is the value of m+bm + b?

  1. A
    22
  2. B
    72\frac{7}{2}
  3. C
    55
  4. D
    77
  5. 1010Answer

Answer

The sum of the slope and the y-intercept of the perpendicular bisector is 1010.
The slope of the segment DEDE is calculated as mDE=4−83−(−5)=−12m_{DE} = \frac{4 - 8}{3 - (-5)} = -\frac{1}{2}. The perpendicular bisector has a slope mm that is the negative reciprocal of this, which is 22. The line must pass through the midpoint of DEDE, which is located at (−5+32,8+42)=(−1,6)\left(\frac{-5 + 3}{2}, \frac{8 + 4}{2}\right) = (-1, 6). Substituting m=2m = 2 and the point (−1,6)(-1, 6) into the slope-intercept form y=mx+by = mx + b yields 6=2(−1)+b6 = 2(-1) + b, which simplifies to b=8b = 8. Summing the slope and the yy-intercept gives m+b=2+8=10m + b = 2 + 8 = 10.

Step-by-Step Solution

1
Find the midpoint of the line segment DEDE.
Midpoint M=(−5+32,8+42)=(−1,6)M = \left(\frac{-5 + 3}{2}, \frac{8 + 4}{2}\right) = (-1, 6)
By definition, the perpendicular bisector must pass through the midpoint of the segment it bisects.
2
Calculate the slope of the line segment DEDE.
mDE=4−83−(−5)=−48=−12m_{DE} = \frac{4 - 8}{3 - (-5)} = \frac{-4}{8} = -\frac{1}{2}
The slope of the segment is needed to find the slope of any line perpendicular to it.
3
Determine the slope mm of the perpendicular bisector.
m=−1mDE=−1−1/2=2m = -\frac{1}{m_{DE}} = -\frac{1}{-1/2} = 2
Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.
4
Find the yy-intercept bb of the perpendicular bisector.
Using the slope-intercept form y=mx+by = mx + b with the slope m=2m = 2 and the midpoint M(−1,6)M(-1, 6): 6=2(−1)+b  ⟹  6=−2+b  ⟹  b=86 = 2(-1) + b \implies 6 = -2 + b \implies b = 8
Substituting a known point on the line allows us to solve for the vertical intercept parameter.
5
Calculate the sum m+bm + b.
m+b=2+8=10m + b = 2 + 8 = 10
This is the value requested by the question.

Key Concept

The perpendicular bisector of a line segment passes through its midpoint at a right angle, meaning its slope is the negative reciprocal of the segment's slope.
Estimated Time:2m 0s
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Topics

Parallel and Perpendicular Lines

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