$ABCD$ is a rhombus. The coordinates of $A$ and $C$ are $\left(3,6\right)$ and $\left(-1,2\right)$ respectively. Write down the equation of $BD$.
Given: $\;$ $ABCD$ is a rhombus with coordinates $A \left(x_1, y_1\right) = \left(3,6\right)$ and $C \left(x_2, y_2\right) = \left(-1,2\right)$
$AC$ is the diagonal of the rhombus $ABCD$
Slope of diagonal $AC = m_1 = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{2 - 6}{-1 - 3} = 1$
The diagonals of a rhombus are perpendicular to each other and bisect each other.
$\therefore \;$ Slope of diagonal $BD = \dfrac{-1}{m_1} = -1$
Midpoint of $AC = M = \left(\dfrac{3 - 1}{2}, \dfrac{6 + 2}{2}\right) = \left(1, 4\right)$
Now, $BD$ also passes through $M \left(1,4\right)$ and has slope $m_2 = -1$
$\therefore \;$ Equation of $BD$ is
$y - 4 = -1 \left(x - 1\right)$
i.e. $\;$ $y - 4 = -x + 1$
i.e. $\;$ $x + y = 5$