Question

The points $A(7,3)$ and $C(0,-4)$ are two opposite vertices of a rhombus $ABCD$, find the equtaion of the diagonal $BD$.

Answer 1

$E$ is the mid pt of $AC$

$\therefore$ Coordinates of pint of $E$ are $(\frac{7+0}{2},\frac{3-4}{2})=(\frac{7}{2},\frac{-1}{2})$

Equation of $AC$ is

$y+4=\frac{3+4}{7-0}(x-7)$

$\Rightarrow 7(y+4)=7(x-7)\Rightarrow 7y+28=7x-49$

$\Rightarrow 7x+7y=28+49=77$

$\Rightarrow x+y=11$

$BD$ is $\perp$ to $AC$

$\therefore$ Equation of $BD$ $=x-y+\lambda =0$

it passes through $E$

$\therefore$ $\frac{7}{2}-\left(\frac{-1}{2}\right)+\lambda =0\Rightarrow \frac{7}{2}+\frac{1}{2}+\lambda =0$

$\Rightarrow \frac{8}{2}+\lambda =0\Rightarrow \lambda =-4$

$\therefore$ Equation of $BD=x-y-4=0\Rightarrow x-y=4$