Question

Prove that the points $(2,-1),(0,2),(2,3)$, and $(4,0)$ are the coordinates of the angular points of a parallelogram and find the angle between its diagonals.

Answer 1

Here with given points, mid point of diagonal is $=(\frac{2+2}{2},\frac{-1+3}{2})=(2,1)$

Now slopes from points $(2,3)$ & $(0,2)$ to $(2,1)$

$\Rightarrow m_2=\frac{3-1}{2-2}=\infty$ & $m_1=\frac{2-1}{0-2}=\frac{-1}{2}$

We know that,

$\tan \theta =\left|\frac{m_2-m_1}{1+m_{2}m1}\right|$

$\Rightarrow tan\theta =\left|\frac{1-\frac{m_1}{m_2}}{\frac{1}{m_2}+m1}\right|$

As $\tan \theta =\left|\frac{1}{0-\frac{1}{2}}\right|$

$\Rightarrow \tan \theta =2$

$\Rightarrow \theta ={\tan }^{-1}\left(2\right)$