Question

If the vertices of a quadrilateral be $A=1+2i,B=-3+i,c=-2-3i$ and $D=2-2i$, then the quadrilateral is a:

• A. Parallelogram
• B. Rectangle
• C. Square
• D. Rhombus

Answer 1

Answer: (C)

Square

Given vertices are $A=1+2i,B=-3+i,C=-2-3i,D=2-2i$

Therefore length of side $AB=\sqrt{(-3-1{)}^2-(1-2{)}^2}=\sqrt{17}$

length of side $BC=\sqrt{(-2+3{)}^2-(-3-1{)}^2}=\sqrt{17}$

length of side $CD=\sqrt{(2+2{)}^2-(-2+3{)}^2}=\sqrt{17}$

length of side $DA=\sqrt{(2-1{)}^2-(-2-2{)}^2}=\sqrt{17}$

Therefore, all sides are equal. Hence, the quadrilateral is either rhombus or square.

The slope of the line $AB=\frac{1-2}{-3-1}=\frac{1}{4}$

The slope of the line $BC=\frac{-3-1}{-2+3}=-4$

The product of the slopes of $AB$ and $BC$ is $\frac{1}{4}\ast -4=-1$

Therefore, the sides are perpendicular to each other.

Hence, the quadrilateral is a square.