Question

if (1,2), (4,3), (x,6) and (3,5) are the vertices of a parallelogram taken in order then find the value of x.

Answer 1

Let (1, 2), (4, 3), (x, 6), and (3, 5) are the coordinates of A, B, C, D vertices of a parallelogram ABCD. We know that diagonals of parallelogram bisect each other.

Therefore, O is the mid-point of AC and BD.

If O is the mid-point of AC, then the coordinates of O are

$\left(\frac{1+\mathrm{x}}{2},\frac{2+6}{2}\right)=\left(\frac{\mathrm{x}+1}{2},4\right)$

If O is the mid-point of BD, then the coordinates of O are

$\left(\frac{4+3}{2},\frac{3+5}{2}\right)=\left(\frac{7}{2},4\right)$

Since both the coordinates are of the same point O,

$$\begin{gathered} \left(\frac{\mathrm{x}+1}{2},4\right)=\left(\frac{7}{2},4\right) \\ \Rightarrow \frac{\mathrm{x}+1}{2}=\frac{7}{2} \\ \Rightarrow \mathrm{x}+1=7 \\ \Rightarrow \mathrm{x}=7-1 \\ \Rightarrow \mathrm{x}=6 \end{gathered}$$