A(7,1) B(6,4) C(5,5) are three points in a place

Question

Find the equations of the perpendicular bisectors of AB and AC
Determine the point of intersection of the perpendicular bisectors in (j)

Answer 1

To find the equations of the perpendicular bisectors of line AB and AC, we first need to find the midpoints of AB and AC, and then find the slopes of the lines AB and AC.

Midpoint of line AB:
x-coordinate: (7+6)/2 = 6.5
y-coordinate: (1+4)/2 = 2.5
Midpoint of AB = (6.5, 2.5)

Midpoint of line AC:
x-coordinate: (7+5)/2 = 6
y-coordinate: (1+5)/2 = 3
Midpoint of AC = (6, 3)

Slope of AB = (4-1)/(6-7) = 3/-1 = -3
Slope of AC = (5-1)/(5-7) = 4/-2 = -2

Now, the equations of the perpendicular bisectors passing through the midpoints of AB and AC will have the negative reciprocal of these slopes. So, the slopes of the perpendicular bisectors will be 1/3 and 1/2 respectively.

Equation of perpendicular bisector of AB:
Using point-slope form, y - 2.5 = 1/3(x - 6.5)
y - 2.5 = 1/3x - 13/3
y = 1/3x - 13/3 + 7/2
y = 1/3x - 13/3 + 21/6
y = 1/3x - 13/3 + 21/6
Equation of perpendicular bisector of AB: y = 1/3x - 13/6

Equation of perpendicular bisector of AC:
Using point-slope form, y - 3 = 1/2(x - 6)
y - 3 = 1/2x - 3
y = 1/2x - 3 + 3
y = 1/2x
Equation of perpendicular bisector of AC: y = 1/2x

To find the point of intersection (j) of the two perpendicular bisectors, we need to solve the two equations simultaneously:

1/3x - 13/6 = 1/2x
1/3x - 1/2x = 13/6
(2-3)x/6 = 13/6
-x/6 = 13/6
x = -13

Substitute x = -13 back into the equation of perpendicular bisector AC to find the y-coordinate:

y = 1/2(-13)
y = -13/2

Therefore, the point of intersection (j) is (-13, -13/2).