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Zorluk: OrtaParallel and Perpendicular Lines

In the standard (x,y)(x, y) coordinate plane, a line segment has endpoints at M(1,−3)M(1, -3) and N(5,5)N(5, 5). If a second line is perpendicular to segment MNMN at its midpoint, what is the yy-intercept of this second line?

Cevap: 2.5

Cevap

The y-intercept of the second line is 2.5.
The slope of segment MNMN is 22, meaning the perpendicular line must have a slope of −0.5-0.5. Since this perpendicular line bisects the segment, it must pass through the midpoint (3,1)(3, 1). Substituting these values into the slope-intercept form y=mx+by = mx + b yields 1=−0.5(3)+b1 = -0.5(3) + b, which simplifies to b=2.5b = 2.5.

Adım Adım Çözüm

1
Calculate the coordinates of the midpoint of segment MNMN.
The midpoint is (3,1)(3, 1).
The perpendicular bisector must pass through the midpoint of the bisected segment.
2
Calculate the slope of segment MNMN.
The slope of MNMN is 22.
The slope is needed to find the perpendicular slope.
3
Calculate the slope of the perpendicular line.
The perpendicular slope is −0.5-0.5.
Perpendicular lines have slopes that are negative reciprocals of each other.
4
Find the yy-intercept (bb) using the slope −0.5-0.5 and the point (3,1)(3, 1).
The yy-intercept is 2.52.5.
Substituting the coordinates of the midpoint and the perpendicular slope into the equation y=mx+by = mx + b gives 1=−0.5(3)+b1 = -0.5(3) + b, which simplifies to b=2.5b = 2.5.

Anahtar Kavram

Finding the equation and y-intercept of a perpendicular bisector using midpoint and negative reciprocal slope.
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Konular

Parallel and Perpendicular Lines

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