Question

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

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

Answer 1

Answer: (B)

rhombus

We first check the lengths of the four sides of the quadrilateral.

$AB=\sqrt{9+16}=\sqrt{25}=5$ units

$BC=\sqrt{(7-3{)}^2+(7-4{)}^2}=\sqrt{9+16}=\sqrt{25}=5$ units

$CD=\sqrt{(4-7{)}^2+(3-7{)}^2}=\sqrt{9+16}=\sqrt{25}=5$ units

$DA=\sqrt{4^2+3^2}=\sqrt{16+9}=\sqrt{25}=5$ units

Since all the sides of the quadrilateral are equal, it can either be a rhombus or a square.

We now check the angle between sides AB and BC.

$\overrightarrow{AB}=3\hat{i}+4\hat{j}$

$\overrightarrow{BC}=4\hat{i}+3\hat{j}$

$cos\theta =\frac{\left(\overrightarrow{AB}\right).\left(\overrightarrow{BC}\right)}{|\overrightarrow{AB}||\overrightarrow{BC}|}=\frac{12+12}{25}=\frac{24}{25}$

Since the angle between two sides is not a right angle, the quadrilateral is a rhombus and not a square.