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Difficulty: MediumCoordinate Geometry: Lines, Slopes, and Distance

In the xyxy-plane, line segment ABAB has endpoints A(−3,1)A(-3, 1) and B(1,9)B(1, 9). Line NN is the perpendicular bisector of segment ABAB. Which of the following statements must be true? Select all that apply.

  1. Line NN has a slope of −12-\frac{1}{2}.Answer
  2. Line NN passes through the point (3,3)(3, 3).Answer
  3. C
    Line NN has a yy-intercept of (0,5)(0, 5).
  4. D
    The length of line segment ABAB is 1010.
  5. E
    The midpoint of line segment ABAB lies in Quadrant I.

Answer

The statement that Line NN has a slope of −12-\frac{1}{2} and the statement that Line NN passes through the point (3,3)(3, 3) are both true.
The slope of segment ABAB is 22, making the perpendicular slope −12-\frac{1}{2}. The midpoint of ABAB is (−1,5)(-1, 5). Using point-slope form with M(−1,5)M(-1, 5) and slope −12-\frac{1}{2}, Line NN has equation y=−12x+92y = -\frac{1}{2}x + \frac{9}{2}. Evaluating at x=3x = 3 gives y=3y = 3, so Line NN passes through (3,3)(3, 3).

Step-by-Step Solution

1
Calculate the slope of line segment ABAB
slope mAB=9−11−(−3)=84=2m_{AB} = \frac{9 - 1}{1 - (-3)} = \frac{8}{4} = 2
The slope of a line passing through (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is y2−y1x2−x1\frac{y_2 - y_1}{x_2 - x_1}.
2
Determine the slope of the perpendicular bisector (Line NN)
slope mN=−1mAB=−12m_N = -\frac{1}{m_{AB}} = -\frac{1}{2}
Perpendicular lines have slopes that are negative reciprocals of each other.
3
Find the midpoint of line segment ABAB
Midpoint M=(−3+12,1+92)=(−1,5)M = \left(\frac{-3 + 1}{2}, \frac{1 + 9}{2}\right) = (-1, 5)
A perpendicular bisector must pass through the midpoint of the segment it bisects.
4
Find the equation of Line NN and test given points
Line NN: y−5=−12(x−(−1))  ⟹  y=−12x+92y - 5 = -\frac{1}{2}(x - (-1)) \implies y = -\frac{1}{2}x + \frac{9}{2}. Testing (3,3)(3, 3): 3=−12(3)+92=62=33 = -\frac{1}{2}(3) + \frac{9}{2} = \frac{6}{2} = 3 (True). Testing yy-intercept (x=0x=0): y=4.5≠5y = 4.5 \neq 5.
Point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) defines the line.
5
Calculate the length of segment ABAB
AB=(1−(−3))2+(9−1)2=42+82=80=45≈8.94AB = \sqrt{(1 - (-3))^2 + (9 - 1)^2} = \sqrt{4^2 + 8^2} = \sqrt{80} = 4\sqrt{5} \approx 8.94
The distance formula between two points is d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.

Key Concept

Coordinate Geometry: Perpendicular Bisectors, Slopes, Midpoints, and Distance
Estimated Time:1m 45s
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Topics

Coordinate Geometry: Lines, Slopes, and Distance

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