Question

If vertices of parallelogram are respectively $(0,0),(1,0),(2,2)\mathrm{and}\left(1,2\right)$ then angle between diagonals is.

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{2}$
• C. $\frac{3\pi }{2}$
• D. $\frac{\pi }{4}$

Answer 1

The correct option is D

$\frac{\pi }{4}$

Explanation for the correct answer:

Finding the angle between the diagonals:

Let, the vertices of the parallelograms $(0,0),(1,0),(2,2)\mathrm{and}\left(1,2\right)$ be $A,B,C\mathrm{and}D$ respectively.

We know, the formula for the slope of a line is $m=\frac{y_2-y_1}{x_2-x_1}=\tan \theta$

Where $x_1,x_2,y_1,y_2$ are the coordinates of the two points.

The slope of the first diagonal $AC=$$\frac{2-0}{2-0}=1$

so, the slope of the first diagonal.

$$\begin{gathered} \tan {\theta }_1=1 \\ \tan {\theta }_1=\tan \left(\frac{\mathrm{\pi }}{4}\right) \\ {\theta }_1={\displaystyle \frac{\mathrm{\pi }}{4}} \end{gathered}$$

The slope of the second diagonal

$BD=$$\frac{2-0}{1-1}=\infty$

so,

$$\begin{gathered} \tan {\theta }_2=\infty \\ \tan {\theta }_2=\tan \left(\frac{\mathrm{\pi }}{2}\right) \\ {\theta }_2=\frac{\mathrm{\pi }}{2} \end{gathered}$$

The angle between the diagonals is

$\begin{array}{rcl}{\theta }_2-{\theta }_1& =& \frac{\mathrm{\pi }}{2}-\frac{\mathrm{\pi }}{4}\\ & =& \frac{\mathrm{\pi }}{4}\end{array}$

Hence, the correct option is D.