If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1,2),(3,4) and…

Question

If in a parallelogram $ABDC$, the coordinates of $A, B$ and $C$ are respectively $(1,2),(3,4)$ and $(2,5)$, then the equation of the diagonal $AD$ is:

Accepted Answer

In parallelogram $ABDC$, diagonals $AD$ and $BC$ bisect each other:
$$\text{midpoint}(AD) = \text{midpoint}(BC) = \left(\frac{3+2}2,\frac{4+5}2\right) = \left(\frac52,\frac92\right)$$

So $D = 2\left(\frac52,\frac92\right)-A = (5,9)-(1,2) = (4,7)$.

Line $AD$ through $(1,2),(4,7)$: slope $=\dfrac53$:
$$y-2=\frac53(x-1) \implies 3y-6=5x-5 \implies 5x-3y+1=0$$

Solutions · 1