Question

The points $A(-4,-1),B(-2,-4),C(4,0)$ and $D(2,3)$ are the vertices of which of the following polygons?

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

Answer 1

Answer: (A), (B)

The correct options are
A parallelogram
B rectangle

First find the distances between these points.

$AB=\sqrt{(-2+4{)}^2+(-4+1{)}^2}=\sqrt{4+9}=\sqrt{13}$

$BC=\sqrt{(4+2{)}^2+(0+4{)}^2}=\sqrt{36+16}=2\sqrt{13}$

$CD=\sqrt{(2-4{)}^2+(3-0{)}^2}=\sqrt{4+9}=\sqrt{13}$

$DA=\sqrt{(-4-2{)}^2+(-1-3{)}^2}=\sqrt{36+16}=2\sqrt{13}$

Thus, we can see that,

$AB=CD$ and $BC=AD$

Now,

Slope of $AB=\frac{-4+1}{-2+4}=-\frac{3}{2}$

Slope of $AD=\frac{3+1}{2+4}=\frac{2}{3}$

Slope of $BC=\frac{0+4}{4+2}=\frac{2}{3}$

Slope of $CD=\frac{3-0}{2-4}=-\frac{3}{2}$

Since,

Slope of $AB\times$ Slope of $AD=-1$

Slope of $AB\times$ Slope of $BC=-1$

Slope of $BC\times$ Slope of $CD=-1$

Slope of $CD\times$ Slope of $AD=-1$

Thus, all the angles are right angles and pair of opposite sides are equal, therefore, the given polygon is a rectangle. It is also to be noted that every rectangle is a special case of a parallelogram. Therefore, both parallelogram and the rectangle are the right choices for the given polygon.