Question

Check whether the points $A(2,-1),B(3,4),C(-2,3)$ and $D(-3,2)$ represents a Rhombus

Answer 1

AB$=\sqrt{(2-3{)}^2+(-1-4{)}^2}$

$=\sqrt{26}$

BC$=\sqrt{(3+2{)}^2+(4-3{)}^2}$

$=\sqrt{26}$

CD$=\sqrt{(-2+3{)}^2+(3+2{)}^2}$

$=\sqrt{26}$

AD$=\sqrt{(2+3{)}^2+(-1+2{)}^2}$

$=\sqrt{26}$

All length of sides are equal

$\therefore$ It can be a square or rhombus

$\therefore$ Length of diagonals.

AC$=\sqrt{(2+2{)}^2+(-1-3{)}^2}$

$=\sqrt{32}$

BD$=\sqrt{(3+3{)}^2+(4+2{)}^2}$

$=\sqrt{72}$

$\therefore$ length of diagonals are not equal

$\therefore$ ABCD is a rhombus not a square.

Hence proved.