Question

Name The Type Of Quadrilateral Formed By The Following Points - $(4,5),(7,6),(4,3),(1,2)$ And Give Reasons For Your Answer

Answer 1

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

Let the points are $A(4,5),B(7,6),S(4,3),D(1,2)$.

Know that the distance $\text{'}d\text{'}$ betwen the two points $\left(x_2-x_1\right)$and $\left(y_2-y_1\right)$is -

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

Find the length of side AB:

$\begin{array}{c}AB=\sqrt{{\left(7-4\right)}^2+{\left(6-5\right)}^2}\\ =\sqrt{3^2+1^2}\\ =\sqrt{9+1}\\ =\sqrt{10}\end{array}$

Step 2: Find the length of side BC.

$\begin{array}{c}BC=\sqrt{{\left(4-7\right)}^2+{\left(3-6\right)}^2}\\ =\sqrt{-{\left(3\right)}^2+3^2}\\ =\sqrt{9+9}\\ =\sqrt{18}\end{array}$

Step 3: Find the length of side CD.

$\begin{array}{c}CD=\sqrt{{\left(1-4\right)}^2+{\left(3-6\right)}^2}\\ =\sqrt{-3^2+{(-1)}^2}\\ =\sqrt{9+1}\\ =\sqrt{10}\end{array}$

Step 4: Find the length of side DA.

$\begin{array}{c}DA=\sqrt{{\left(1-4\right)}^2+{\left(2-5\right)}^2}\\ =\sqrt{{(-3)}^2+{(-3)}^2}\\ =\sqrt{9+9}\\ =\sqrt{18}\end{array}$

Step 5: Find the length of diagonal AC.

$\begin{array}{c}AC=\sqrt{{\left(4-4\right)}^2+{\left(3-5\right)}^2}\\ =\sqrt{0^2+{(-2)}^2}\\ =\sqrt{0+4}\\ =2\end{array}$

Step 6: Find the length of diagonal BD.

$\begin{array}{c}BD=\sqrt{{\left(1-7\right)}^2+{\left(2-6\right)}^2}\\ =\sqrt{{(-6)}^2+{(-4)}^2}\\ =\sqrt{36+16}\\ =\sqrt{52}\end{array}$

Since side AB is equal to CD and side BC is equal to AD i.e. the opposite sides of this quadrilateral are equal but the diagonals are not equal.

Hence the quadrilateral formed by joining the points $(4,5),(7,6),(4,3),(1,2)$is a parallelogram.