Write an equation of the perpendicular bisector of JK, where J = (−4, 2) and K = (2, 2).
=__________
+2446 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.
