In a rhombus ABCD the diagonals AC and BD intersect at the point (3, 4). If the point A is (1, 2) then the diagonal BD has the equation

P = (3, 4) and A = (1, 2)

Slope of $\overline{AP}=\frac{2-4}{1-3}=\frac{-2}{-2}=1$

$\begin{array}{l}\overline{AP}\perp \overline{BD}\Rightarrow Slopeof\overline{BD}=\frac{-1}{SlopeofAP}\\ m=\frac{-1}{1}=-1\\ Equationof\overline{BD}isy-y_1=m\left(x-x_1\right)\\ \Rightarrow y-4=-1\left(x-3\right)\\ \Rightarrow y-4=-x+3\\ \Rightarrow x+y-7=0\end{array}$