Points A and B have coordinates (8, 3) and (p, q) respectively. The equation of the perpendicular
bisector of AB is y = -2x +4.
Find the values of p and q.

Answer:

p = -4, q = -3

Step-by-step explanation:

y = -2x +4  ... (1)   perpendicular bisector of AB, slope = -2

slope of AB = 1/2

Line AB pass (8,3): (y-3) / (x-8) = 1/2

AB equation: y-3 = 1/2(x-8)      y = 1/2x - 1  ... (2)

(2)-(1): 5/2 x = 5      x = 2

y = 0       (2,0) intercept of bisector and AB, it is midpoint of A (8,3) and (p,q)

(8+p)/2 = 2

p = -4

(3+q)/2 = 0

q = -3

The values of p and q from the perpendicular bisector are;

p = -4 and q = -3

We are given coordinates of A and B as;

A(8, 3) and B(p, q)

Bisector of AB is y = -2x + 4

Thus, from y = mx + c, Slope of the bisector is -2.

Since the bisector is perpendicular to AB, then slope of AB = -(1/-2) = 1/2

Thus, slope of the line from point A to the point of the bisector on AB is;

(y - 3)/(x - 8) = 1/2

y - 3 = ¹/₂(x - 8)

y - 3 = ¹/₂x - 4

y = ¹/₂x - 1

Putting ¹/₂x - 1 for y in the bisector equation gives;

¹/₂x - 1 = -2x + 4

2x + ¹/₂x = 4 + 1

2.5x = 5

x = 5/2.5

x = 2

Thus;

y = -2(2) + 4

y = 0

Thus, the coordinate of the midpoint of AB is (2, 0)

Now, this means that;

(8 + p)/2 = 2

8 + p = 4

p = 4 - 8

p = 4

Also; (3 + q)/2 = 0

3 + q = 0

q = -3

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