A segment bisector contains the midpoint of the segment. We will use the Midpoint Formula to find the midpoint M of DE.
M( x_1+x_2/2,y_1+y_2/2 )To find the coordinates of the midpoint M, we will substitute (- 2,- 4) and (2,1) for (x_1,y_1) and (x_2,y_2) in the formula.
A perpendicular bisector is perpendicular to the segment through the midpoint. In order to find the slope of the bisector, we will first find the slope of DE. To do so, we will use the Slope Formula.
m = y_2-y_1/x_2-x_1
Let's substitute (- 2,- 4) and (2,1) for (x_1,y_1) and (x_2,y_2) in this formula.
The slope of DE is 54. Let m_p be the slope of the perpendicular bisector. The product of the slopes of two perpendicular lines is - 1.
5/4 * m_p = - 1 ⇔ m_p = - 4/5
The slope of the perpendicular bisector of DE is - 45.
Equation of the Perpendicular Bisector
Since we know a point and the slope of the bisector, we will use the point-slope form of a line to write its equation.
y-y_1=m(x-x_1)
Let's substitute - 45 for m and ( 0, - 32 ) for (x_1,y_1) in the formula.
