Question

Three points have coordinates A(2,6), B(8,10), and C(6,0). The perpendicular bisector of AB meets the line BC at D. Find:

(i) The equation of the perpendicular bisector of AB in the form (ax + by = C).
(ii) The coordinates of D.
a) (3x - y = 15), D(5, 2)
b) (2x + 3y = 20), D(7, 5)
c) (x - 2y = 5), D(4, 3)
d) (4x + y = 10), D(6, 1)

Answer 1

The equation of the perpendicular bisector of AB is 3x + 2y = 31. Point D does not exist.

Step-by-step explanation:

To find the equation of the perpendicular bisector of AB, we first find the midpoint of AB using the midpoint formula: (x, y) = ((x1 + x2) / 2, (y1 + y2) / 2). Plugging in the coordinates of A(2, 6) and B(8, 10), we get the midpoint M(5, 8). The slope of AB is (10 - 6) / (8 - 2) = 4 / 6 = 2 / 3. The slope of the perpendicular bisector is the negative reciprocal of the slope of AB, so it is -3 / 2. Using the point-slope form y - y1 = m(x - x1), we can write the equation of the perpendicular bisector as y - 8 = (-3 / 2)(x - 5). Simplifying, we get 2y - 16 = -3x + 15, or 3x + 2y = 31.

To find the coordinates of point D, we solve the system of equations formed by the equation of the perpendicular bisector and the equation of BC. Substituting the equation of the perpendicular bisector into the equation of BC, we get 3x + 2y = 31. Plugging in the coordinates of C(6, 0), we get 3(6) + 2(0) = 31, which gives us 18 = 31. This is not true, so there is no solution to the system of equations. Therefore, D does not exist.