How do you find the equation of the perpendicular bisector of the segment joining the points A #(6, -3)# and B #(-2, 5)#?

Question

Evelyn Arboleda · Accepted Answer

#"The eqn. of p.b. is "x-y-1=0.#

We know from geometry that a surface's perpendicular bisector (p.b.)

The line that passes through its midpoint (m.p.) is known as a line segment (sgmt.) &

with respect to the sgmt.

Let us denote, by #bar(AB),# the line sgmt. joining the pts.

#A(6,-3), and, B(-2,5).#

#M# is the m.p. of #bar(AB)#

#rArr M=M((6-2)/2,(-3+5)/2)=M(2,1).#

Slope of #bar(AB)=(5-(-3))/(-2-6)=8/-8=-1.#

#:." The Slope of p.b. of "bar(AB)" must be 1."#

We obtain its eqn by using the Slope-Point Form of line for the p.b.

#y-1=1(x-2) rArr x-y-1=0.#

Procedure II:=

We make use of the following p.b. of a line sgmt. geometrical property.

#"Any pt. on the p.b. of a line sgmt. is equidistant from the "#

#"end-points of the sgmt."#

Let #P(x,y)# be any pt. on the p.b. of #bar(AB).#

#"Then, dist. PA= dist. PB" rArr PA^2=PB^2#.

#rArr (x-6)^2+(y+3)^2=(x+2)^2+(y-5)^2#

#rArr x^2-12x+36+y^2+6y+9#

#=x^2+4x+4+y^2-10y+25#

#rArr -16x+16y+16=0#

#rArr x-y-1=0.#, as before!

Have fun with math!

Jayden Jackson · Answer

undefined To find the equation of the perpendicular bisector of the segment joining the points $ A(6, -3) $ and $ B(-2, 5) $, follow these steps:

1. **Find the midpoint** of the segment joining $ A $ and $ B $:
$$ \text{Midpoint} \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$

Using the coordinates of $ A $ and $ B $:
$$ \text{Midpoint} \left( \frac{6 + (-2)}{2}, \frac{(-3) + 5}{2} \right) $$
$$ \text{Midpoint} \left( 2, 1 \right) $$

2. **Find the slope** of the line passing through $ A $ and $ B $:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

Using the coordinates of $ A $ and $ B $:
$$ m = \frac{5 - (-3)}{-2 - 6} $$
$$ m = \frac{8}{-8} $$
$$ m = -1 $$

The negative reciprocal of the slope $ m $ is the slope of the perpendicular bisector.

$$ m_{\text{perpendicular}} = \frac{-1}{m} $$
$$ m_{\text{perpendicular}} = \frac{-1}{-1} $$
$$ m_{\text{perpendicular}} = 1 $$

3. **Use the point-slope form** to find the equation of the perpendicular bisector:
$$ y - y_1 = m_{\text{perpendicular}}(x - x_1) $$

Using the midpoint $ (2, 1) $:
$$ y - 1 = 1(x - 2) $$
$$ y - 1 = x - 2 $$
$$ y = x - 1 $$

So, the equation of the perpendicular bisector of the segment joining $ A(6, -3) $ and $ B(-2, 5) $ is $ y = x - 1 $.