Question

Show that the points $A(2,-2),B(8,4),C(5,7)$ and $D(-1,1)$ are the angular points of a rectangle.

Answer 1

Let $ABCD$ be the quadrilateral. [To show $ABCD$ is a rectangle]

Where, $A(2,-2),B(8,4),C(5,7)$ and $D(-1,1)$

using distance fromula,

$AB=\sqrt{(8-2{)}^2+(4-(-2){)}^2}$ units

$AB=\sqrt{36+36}$

$AB=\sqrt{72}$ units...(1)

$BC=\sqrt{(5-8{)}^2+(7-4{)}^2}$

$BC=\sqrt{9+9}=\sqrt{18}$ units...(2)

$CD=\sqrt{(-1-5{)}^2+(1-7{)}^2}$

$CD=\sqrt{36+36}=\sqrt{72}$ units...(3)

$DA=\sqrt{(2-(-1){)}^2+(-2-1{)}^2}$

$DA=\sqrt{18}$ units...(4)

$AC=\sqrt{(5-2{)}^2+(7-(-2){)}^2}$

$4AC=\sqrt{9+81}=\sqrt{90}$ units...(5)

$BD=\sqrt{(-1-8{)}^2+(1-4{)}^2}$

$BD=\sqrt{90}$ units ...(6)

from (1), (2), (3), (4) we get

$AB=CD,BC=AD$ ...(7)

from (5) and (6), we get

$AC=BD$...(8)

$7\Rightarrow$ Opposite side of quadrilateral are equal

$8\Rightarrow$ Daigonal of quadrilateral are equal

$\therefore ABCD$ is rectangle.