Question

Given A(2,0) and B(8,4), show that P(3,5) is on the perpendicular bisector of line AB by these 2 methods

1)Show line AB is perpendicular to PM where M is the midpoint of line AB
2) Show PA is equal to PB

Answer 1

Gradient of AB is 4 - 0 / 8 - 2 = 4/6 = 2/3

Gradient of PM is -1 divide by 2/3 = -3/2

Solution for (1):

The midpoint of AB , M, is (2,0) + 1/2(8 - 2, 4 - 0)

(2,0) + (3,2) = 5,2

The gradient of PM then is (5 - 2) / (3 - 5) = -3/2

Therefore AB is perpendicular to PM since gradient of two perpendicular bisectors is -1

Solution for (2):

AM^2 + PM^2 = PA^2 , PM^2 + MB^2 = PB^2

AM^2 + PM^2 must be equal to PM^2 + MB^2

and since PM passes through the middle of AB then;

PA^2 = PB^2

(Finding the square root)

PA = PB