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# Write an equation of the perpendicular bisector of JK

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Write an equation of the perpendicular bisector of JK, where J = (−4, 2) and K = (2, 2).

=__________

Sep 28, 2018

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Slope-Intercept or Point-Slope?

Sep 28, 2018
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A perpendicular bisector is a line that intersects a segment at a right angle and at that segment's midpoint. In order to generate a unique equation of any line, you need to know at least one point and the slope; without these, it is impossible to generate a specific equation.

The definition I provided should give a clue as to how to find one point of the perpendicular bisector. Since we know that perpendicular bisectors intersect a segment at its midpoint, we can find the midpoint of $$\overline{JK}$$ :

 $$m_{\overline{JK}}=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$$ This is the midpoint formula. The midpoint formula just finds the average of the two coordinates. $$m_{\overline{JK}}=\left(\frac{-4+2}{2},\frac{2+2}{2}\right)$$ The only thing left to do is to simplify. $$m_{\overline{JK}}=(-1,2)$$ Yes! We have already found one part of the perpendicular bisector, a point.

Finding the slope requires a slightly indirect approach. We can determine the slope of the given segment and adjust with the knowledge of perpendicular lines. I need not use the slope formula here because the y-coordinates of both points are the same, which means that the segment that connects J and K is horizontal. All horizontal lines have a slope of 0. The line of the perpendicular bisector, therefore, is vertical because a vertical line is perpendicular to a horizontal line. A vertical line has an undefined slope. Look at that! We have determined the slope of the perpendicular bisector.

Vertical lines are written in the form of x=c, where c is a constant. We know, from the information gathered from solving for the midpoint, that $$x=-1$$ is the equation of the perpendicular bisector.

Sep 28, 2018