Answer:
Each of the points and the y-intercept of their perpendicular bisectors
1) A(-4,5) and B(8,9), y-intercept = 13
2) A(2, 4) and B(-8,6), y-intercept = 20
3) A(5, 4) and B(7.2), y-intercept = -3
4) A(2, 9) and B(-4.3), y-intercept = 5
5) A(3.-2) and B(9.-12), y-intercept = -10.6
6) A(4, 10) and B(8, 12), y-intercept = 23
Arranged in order of increasing y-intercepts of their perpendicular bisectors, from the smallest to largest y-intercept
5) A(3.-2) and B(9.-12), y-intercept = -10.6
3) A(5, 4) and B(7.2), y-intercept = -3
4) A(2, 9) and B(-4.3), y-intercept = 5
1) A(-4,5) and B(8,9), y-intercept = 13
2) A(2, 4) and B(-8,6), y-intercept = 20
6) A(4, 10) and B(8, 12), y-intercept = 23
Step-by-step explanation:
The slopes of two perpendicular lines are related as thus
m₁m₂ = -1
Hence, for each of the two Points given, the slope of the perpendicular bisector is
m₂ = -(1/m₁)
But the slope of each of the lines connecting the two points is given as
m = (y₁ - y₂)/(x₁ - x₂)
And the coordinates of the midpoint, that the perpendicular bisector passes through is given as
(x, y) = {[(x₁ + x₂)/2], [(y₁ + y₂)/2]}
And from the slope of the perpendicular bisector and the coordinates of the midpoint of each question point, we can obtain the equation of the line that is the perpendicular bisector. And easily obtain the y-intercept from that.
Taking the points, one at time
1) A(-4,5) and B(8,9)
Slope of the line connecting the two points = m₁ = (9 - 5)/(8 - -4) = (4/12) = (1/3)
Slope of the perpendicular bisector
= m₂ = -1 ÷ (1/3) = -3
The midpoint of the two points is given as
= [(-4 + 8)/2, (9 + 5)/2]
= (2, 7)
The equation of the perpendicular bisector is then given as the equation of line with slope -3 and passes through (2, 7)
y = mx + c
7 = (-3×2) + c
7 = -6 + c
c = 7 + 6 = 13
y = -3x + 13
y-intercept = 13
2) A(2, 4) and B(-8,6)
Slope of the line connecting the two points = m₁ = (6 - 4)/(-8 - 2) = -(2/10) = -(1/5)
Slope of the perpendicular bisector
= m₂ = -1 ÷ -(1/5) = 5
The midpoint of the two points is given as
= [(2 + -8)/2, (4 + 6)/2]
= (-3, 5)
The equation of the perpendicular bisector is then given as the equation of line with slope 5 and passes through (-3, 5)
y = mx + c
5 = (5×-3) + c
5 = -15 + c
c = 5 + 15 = 20
y = 5x + 20
y-intercept = 20
3) A(5, 4) and B(7.2)
Slope of the line connecting the two points = m₁ = (2 - 4)/(7 - 5) = -(2/2) = -1
Slope of the perpendicular bisector
= m₂ = -1 ÷ -1 = 1
The midpoint of the two points is given as
= [(5 + 7)/2, (4 + 2)/2]
= (6, 3)
The equation of the perpendicular bisector is then given as the equation of line with slope 1 and passes through (6, 3)
y = mx + c
3 = (1×6) + c
3 = 6 + c
c = 3 - 6 = -3
y = x - 3
y-intercept = -3
4) A(2, 9) and B(-4.3)
Slope of the line connecting the two points = m₁ = (3 - 9)/(-4 - 2) = (-6/-6) = 1
Slope of the perpendicular bisector
= m₂ = -1 ÷ 1 = -1
The midpoint of the two points is given as
= [(2 + -4)/2, (9 + 3)/2]
= (-1, 6)
The equation of the perpendicular bisector is then given as the equation of line with slope -1 and passes through (-1, 6)
y = mx + c
6 = (-1×-1) + c
6 = 1 + c
c = 6 - 1 = 5
y = -x + 5
y-intercept = 5
5) A(3.-2) and B(9.-12)
Slope of the line connecting the two points = m₁ = (-12 - -2)/(9 - 3) = (-10/6) = -(5/3)
Slope of the perpendicular bisector
= m₂ = -1 ÷ (-5/3) = (3/5)
The midpoint of the two points is given as
= [(3 + 9)/2, (-2 + -12)/2]
= (6, -7)
The equation of the perpendicular bisector is then given as the equation of line with slope 3/5 and passes through (6, -7)
y = mx + c
-7 = [(3/5)×6] + c
-7 = 3.6 + c
c = -7 + -3.6 = -10.6
y = 3x/5 - 10.6
y-intercept = -10.6
6) A(4, 10) and B(8, 12)
Slope of the line connecting the two points = m₁ = (12 - 10)/(8 - 4) = (2/4) = (1/2)
Slope of the perpendicular bisector
= m₂ = -1 ÷ (1/2) = -2
The midpoint of the two points is given as
= [(4 + 8)/2, (10 + 12)/2]
= (6, 11)
The equation of the perpendicular bisector is then given as the equation of line with slope -2 and passes through (6, 11)
y = mx + c
11 = (-2×6) + c
11 = -12 + c
c = 11 + 12 = 23
y = -2x + 23
y-intercept = 23
1) A(-4,5) and B(8,9), y-intercept = 13
2) A(2, 4) and B(-8,6), y-intercept = 20
3) A(5, 4) and B(7.2), y-intercept = -3
4) A(2, 9) and B(-4.3), y-intercept = 5
5) A(3.-2) and B(9.-12), y-intercept = -10.6
6) A(4, 10) and B(8, 12), y-intercept = 23
Hope this Helps!!!
