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Difficulty: MediumParallel and Perpendicular Lines

In the standard (x,y)(x, y) coordinate plane, what is the yy-intercept of the perpendicular bisector of the line segment with endpoints (0,1)(0, 1) and (2,5)(2, 5)?

  1. A
    43\frac{4}{3}
  2. B
    52\frac{5}{2}
  3. 72\frac{7}{2}Answer
  4. D
    55
  5. E
    −52-\frac{5}{2}

Answer

72\frac{7}{2}
To find the perpendicular bisector, we first locate the midpoint of the segment with endpoints (0,1)(0, 1) and (2,5)(2, 5), which is (0+22,1+52)=(1,3)\left(\frac{0 + 2}{2}, \frac{1 + 5}{2}\right) = (1, 3). The slope of this segment is m=5−12−0=2m = \frac{5 - 1}{2 - 0} = 2. The slope of the perpendicular bisector is the negative reciprocal, −12-\frac{1}{2}. Using the point-slope form with the midpoint (1,3)(1, 3) and slope −12-\frac{1}{2}, the equation of the perpendicular bisector is y−3=−12(x−1)y - 3 = -\frac{1}{2}(x - 1). Setting x=0x = 0 to find the yy-intercept gives y=3+12=72y = 3 + \frac{1}{2} = \frac{7}{2}.

Step-by-Step Solution

1
Find the midpoint of the line segment with endpoints (0,1)(0, 1) and (2,5)(2, 5).
The midpoint is M=(0+22,1+52)=(1,3)M = \left(\frac{0 + 2}{2}, \frac{1 + 5}{2}\right) = (1, 3).
A perpendicular bisector must pass through the midpoint of the segment it bisects.
2
Calculate the slope of the original line segment.
The slope is m=5−12−0=42=2m = \frac{5 - 1}{2 - 0} = \frac{4}{2} = 2.
The slope of the segment is needed to find the slope of the line perpendicular to it.
3
Determine the slope of the perpendicular bisector.
The perpendicular slope is m⊥=−1m=−12m_{\perp} = -\frac{1}{m} = -\frac{1}{2}.
The slope of a perpendicular line is the negative reciprocal of the original line's slope.
4
Write the equation of the perpendicular bisector and find the yy-intercept.
Using the point-slope form with point (1,3)(1, 3) and slope −12-\frac{1}{2} gives the equation y−3=−12(x−1)y - 3 = -\frac{1}{2}(x - 1). Setting x=0x = 0 to find the yy-intercept yields y−3=−12(0−1)=12y - 3 = -\frac{1}{2}(0 - 1) = \frac{1}{2}, which simplifies to y=3+12=72y = 3 + \frac{1}{2} = \frac{7}{2}.
The yy-intercept is the value of the function when x=0x = 0.

Key Concept

Perpendicular Bisectors in the Coordinate Plane
Estimated Time:1m 30s
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Topics

Parallel and Perpendicular Lines

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