A(7, -8) and C(1, 4) are vertices of a square ABCD. Find equation of diagonal BD?

Solution (By Examveda Team)

By the midpoint formula, $$\left( {\frac{{{x_1} + {x_2}}}{2},\,\frac{{{y_1} + {y_2}}}{2}} \right)$$
Midpoint of line AC $$ = \left( {\frac{{7 + 1}}{2},\,\frac{{ - 8 + 4}}{2}} \right) = \left( {4,\, - 2} \right)$$
O(x₃, y₃) = (4, -2)
(M₁) Slope of line AC,
$${M_1} = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \frac{{4 - \left( { - 8} \right)}}{{1 - 7}} = - 2$$
If the lines are ⊥, then M₁ × M₂ = -1
- 2 × M₂ = -1
M₂ = $$\frac{1}{2}$$
∴ Equation of line BD
y - y₃ = M₂(x - x₃)
y - (-2) = $$\frac{1}{2}$$(x - 4)
y + 2 = $$\frac{1}{2}$$(x - 4)
x - 2y = 8