ABCD is a parallelogram where A(x, y), B(5, 8), C(4, 7) and D(2, -4). Find

(i) the coordinates of A.

(ii) the equation of the diagonal BD.

(i) Let O be the point of intersection of the diagonals.

So, O will be the mid-point of the diagonals so also the mid-point of BD.

By mid-point formula coordinates of O are,

$= \Big(\dfrac{5 + 2}{2}, \dfrac{8 - 4}{2}\Big) \\[1em] = \Big(\dfrac{7}{2}, \dfrac{4}{2}\Big) \\[1em] = \Big(\dfrac{7}{2}, 2\Big).$

Since, O is also the mid-point of AC so,

$\Rightarrow \dfrac{7}{2} = \dfrac{x + 4}{2} \text{ and } 2 = \dfrac{y + 7}{2} \\[1em] \Rightarrow 7 = x + 4 \text{ and } 4 = y + 7 \\[1em] \Rightarrow x = 7 - 4 \text{ and } y = 4 - 7 \\[1em] \Rightarrow x = 3 \text{ and } y = -3.$

Hence, the coordinates of A are (3, -3).

(ii) Equation of diagonal BD can be given by two point formula,

$y - y$1 = \dfrac{y2 - y1}{x2 - x1}(x - x1)

Putting values in above equation,

$\Rightarrow y - 8 = \dfrac{-4 - 8}{2 - 5}(x - 5) \\[1em] \Rightarrow y - 8 = \dfrac{-12}{-3}(x - 5) \\[1em] \Rightarrow y - 8 = 4(x - 5) \\[1em] \Rightarrow y - 8 = 4x - 20 \\[1em] \Rightarrow 4x - y - 20 + 8 = 0 \\[1em] \Rightarrow 4x - y - 12 = 0.$

Hence, the equation of the diagonal BD is 4x - y - 12 = 0.