Question

Name the quadrilateral formed, if any by the following points, and give reaons for your answer.
(i) $(-1,-2)$, $(1,0)$, $(-1,2)$, $(-3,0)$
(ii)$(-3,5)$, $(3,1)$, $(0,3)$, $(-1,-4)$
(iii)$(4,5)$, $(7,6)$, $(4,3)$, $(1,2)$

Answer 1

Given, $A(-1,-2)$, $B(1,0)$, $C(-1,2)$, $D(-3,0)$ Determine distances: $AB$, $BC$, $CD$, $DA$, $AC$ and $BD$.
$AB=\sqrt{(1+1{)}^2+(0+2{)}^2}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}$
$BC=\sqrt{(-1-1{)}^2+(2-0{)}^2}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}$
$CD=\sqrt{(-3+1{)}^2+(0-2{)}^2}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}$
$DA=\sqrt{(-1+3{)}^2+(-2-0{)}^2}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}$ $AB=BC=CD=DA$
Since, the sides of the quadrilateral are equal.
$AC=\sqrt{(-1+1{)}^2+(2+2{)}^2}=\sqrt{0+16}=4$ $BD=\sqrt{(-3-1{)}^2+(0-0{)}^2}=\sqrt{16+0}=4$
Diagonals of the quadrilateral are also equal.
$\therefore$
we conclude that $ABCD$ is a square.