Find the coordinates of the mid-points of the line segments joining the following pairs of points :

(i) (2, -3), (-6, 7)

(ii) (5, -11), (4, 3)

(iii) (a + 3, 5b), (2a - 1, 3b + 4).

(i) We know that,

Coordinates of mid-point of pair of points (x₁, y₁) and (x₂, y₂) is given by,

$\Rightarrow \Big(\dfrac{x$1 + x2}{2}, \dfrac{y1 + y2}{2}\Big)

∴ Mid-point of (2, -3), (-6, 7) is,

$\Rightarrow \Big(\dfrac{2 + (-6)}{2}, \dfrac{(-3) + (7)}{2}\Big) \\[1em] = \Big(-\dfrac{4}{2}, \dfrac{4}{2}\Big) \\[1em] = (-2, 2).$

Hence, (-2, 2) is the mid-point of (2, -3) and (-6, 7).

(ii) We know that,

Coordinates of mid-point of pair of points (x₁, y₁) and (x₂, y₂) is given by,

$\Rightarrow \Big(\dfrac{x$1 + x2}{2}, \dfrac{y1 + y2}{2}\Big)

∴ Mid-point of (5, -11), (4, 3) is,

$\Rightarrow \Big(\dfrac{5 + 4}{2}, \dfrac{(-11) + 3}{2}\Big) \\[1em] = \Big(\dfrac{9}{2}, -\dfrac{8}{2}\Big) \\[1em] = \Big(\dfrac{9}{2}, -4\Big).$

Hence, $(\dfrac{9}{2}, -4)$ is the mid-point of (5, -11) and (4, 3).

(iii) We know that,

Coordinates of mid-point of pair of points (x₁, y₁) and (x₂, y₂) is given by,

$\Rightarrow \Big(\dfrac{x$1 + x2}{2}, \dfrac{y1 + y2}{2}\Big)

∴ Mid-point of (a + 3, 5b), (2a - 1, 3b + 4) is,

$\Rightarrow \Big(\dfrac{a + 3 + 2a - 1}{2}, \dfrac{5b + 3b + 4}{2}\Big) \\[1em] = \Big(\dfrac{3a + 2}{2}, \dfrac{8b + 4}{2}\Big) \\[1em] = \Big(\dfrac{3a + 2}{2}, 4b + 2\Big).$

Hence, $(\dfrac{3a + 2}{2}, 4b + 2)$ is the mid-point of (a + 3, 5b) and (2a - 1, 3b + 4).