Answer:
Part 1) The equation of the perpendicular bisector of the segment AB is
[tex]y-1=2(x-1)[/tex] or [tex]y=2x-1[/tex]
Part 2) [tex]dPA=2.5\ units[/tex]
Part 3) [tex]dPB=2.5\ units[/tex]
Step-by-step explanation:
step 1
Find the midpoint AB (because a bisector divide into two equal parts)
we have
A(3, 0) and B(–1, 2)
[tex]M=(\frac{3-1}{2},\frac{0+2}{2})[/tex]
[tex]M=(1,1)[/tex]
step 2
Find the slope AB
we have
A(3, 0) and B(–1, 2)
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
substitute the values
[tex]mAB=\frac{2-0}{-1-3}[/tex]
[tex]mAB=\frac{2}{-4}[/tex]
[tex]mAB=-\frac{1}{2}[/tex]
step 3
Find the slope of the perpendicular bisector of the segment AB
Remember that
If two lines are perpendicular, then their slopes are opposite reciprocal (the product is equal to -1)
m1*m2=-1
we have
[tex]m1=-\frac{1}{2}[/tex]
so
[tex]m2=2[/tex]
step 4
Find the equation of the perpendicular bisector of the segment AB
[tex]y-y1=m(x-x1)[/tex]
we have
[tex]m=2[/tex]
[tex](x1,y1)=(1,1)[/tex] -----> midpoint AB
substitute
[tex]y-1=2(x-1)[/tex]
Convert to slope intercept form
[tex]y=2x-2+1\\y=2x-1[/tex]
step 5
Find the x-intercept (point P)
For y=0
0=2x-1
2x=1
x=0.5
The x-intercept is the point P(0.5,0)
step 6
Find the length PA
we have
P(0.5,0),A(3,0)
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
substitute the values
[tex]d=\sqrt{(0-0)^{2}+(3-0.5)^{2}}[/tex]
[tex]d=\sqrt{(0)^{2}+(2.5)^{2}}[/tex]
[tex]dPA=2.5\ units[/tex]
step 7
Find the length PB
we have
P(0.5,0),B(-1,2)
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
substitute the values
[tex]d=\sqrt{(2-0)^{2}+(-1-0.5)^{2}}[/tex]
[tex]d=\sqrt{(2)^{2}+(-1.5)^{2}}[/tex]
[tex]dPB=2.5\ units[/tex]
