Given M is the mid-point of AB, find the co-ordinates of:

(i) A; if M = (1, 7) and B = (-5, 10),

(ii) B; if A = (3, -1) and M = (-1, 3).

By formula,

Mid-point (M) = $\Big(\dfrac{x$1 + x2}{2}, \dfrac{y1 + y2}{2}\Big)

(i) Let co-ordinates of A = (x, y).

Substituting values in above formula we get,

$(1, 7) = \Big[\dfrac{x + (-5)}{2}, \dfrac{y + 10}{2}\Big] \\[1em] \therefore 1 = \dfrac{x - 5}{2} \text{ and } 7 = \dfrac{y + 10}{2} \\[1em] \Rightarrow 2 = x - 5 \text{ and } 14 = y + 10 \\[1em] \Rightarrow x = 7 \text{ and } y = 4.$

A = (x, y) = (7, 4).

Hence, co-ordinates of A = (7, 4).

(ii) Let co-ordinates of B = (x, y).

Substituting values in above formula we get,

$(-1, 3) = \Big(\dfrac{3 + x}{2}, \dfrac{-1 + y}{2}\Big) \\[1em] \therefore -1 = \dfrac{3 + x}{2} \text{ and } 3 = \dfrac{y - 1}{2} \\[1em] \Rightarrow -2 = x + 3 \text{ and } 6 = y - 1 \\[1em] \Rightarrow x = -5 \text{ and } y = 7.$

B = (x, y) = (-5, 7).

Hence, co-ordinates of B = (-5, 7).