Question

Name The Type Of Quadrilateral Formed By The Following Points -$(-1,-2),(1,0),(-1,2),(-3,0)$ And Give Reasons For Your Answer

Answer 1

Step 1. Find the distance between the points using the distance formula.

Let the points are $A(-1,-2),B(1,0),C(-1,2),D(-3,0)$.

Know that the distance$\left(d\right)$ between the points $\left(x_1,y_1\right)$and $\left(x_2,y_2\right)$ is-

$d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Find the length of side AB:

$\begin{array}{c}AB=\sqrt{{\left(1+1\right)}^2+{\left(0-2\right)}^2}\\ =\sqrt{4+4}\\ =\sqrt{8}\\ =2\sqrt{2}\end{array}$

Step 2: Find the length of side BC:

$\begin{array}{c}BC=\sqrt{{\left(-1-1\right)}^2+{\left(2-0\right)}^2}\\ =\sqrt{4+4}\\ =\sqrt{8}\\ =2\sqrt{2}\end{array}$

Step 3: Find the length of side CD:

$\begin{array}{c}CD=\sqrt{{\left(-3+1\right)}^2+{\left(0-2\right)}^2}\\ =\sqrt{4+4}\\ =\sqrt{8}\\ =2\sqrt{2}\end{array}$

Step 4: Find the length of side DA:

$\begin{array}{c}DA=\sqrt{{\left(-3+1\right)}^2+{\left(0+2\right)}^2}\\ =\sqrt{4+4}\\ =\sqrt{8}\\ =2\sqrt{2}\end{array}$

Step 5: Find the length of diagonal AC:

$\begin{array}{c}AC=\sqrt{{\left(-1+1\right)}^2+{\left(2+2\right)}^2}\\ =\sqrt{0+16}\\ =\sqrt{16}\\ =4\end{array}$

Step 6: Find the length of diagonal BD:

$\begin{array}{c}BD=\sqrt{{\left(-3-1\right)}^2+{\left(0+0\right)}^2}\\ =\sqrt{16+0}\\ =\sqrt{16}\\ =4\end{array}$

Since All sides of the quadrilateral are equal and diagonals are also equal.

Hence the Quadrilateral Formed By joining the Points -$(-1,-2),(1,0),(-1,2),(-3,0)$ is a square.