Question

The quadrilateral $ABCD$ formed by the points $A(0,0),B(3,4),C(7,7)$ and $D(4,3)$ is a

• A. rectangle
• B. rhombus
• C. square
• D. parallelogram

Answer 1

The correct option is B rhombus

The sides of the quadrilateral
$AB=\sqrt{(3-0{)}^2+(4-0{)}^2}=5$
$BC=\sqrt{(7-3{)}^2+(7-4{)}^2}=5$
$CD=\sqrt{(4-7{)}^2+(3-7{)}^2}=5$
$DA=\sqrt{(0-4{)}^2+(0-3{)}^2}=5$
Hence, $AB=BC=CD=DA$, It may be square or rhombus
Now, check the diagonals
$AC=\sqrt{(7-0{)}^2+(7-0{)}^2}=7\sqrt{2}$
$BD=\sqrt{(4-3{)}^2+(3-4{)}^2}=\sqrt{2}$
Here, $AC\ne BD$
Thus, quadrilateral $ABCD$ is rhombus.