Question

Examine whether the following points taken in order form a rectangle.

(-3, 0), (1, -2), (5, 6) and (1, 8)

Answer 1

Let us consider the given points as $A(-3,0);B(1,-2);C(5,6);D(1,8)$.

The distance between $AB=\sqrt{(-3-1{)}^2+(0+2{)}^2}=\sqrt{16+4}=\sqrt{20}$

The distance between $BC=\sqrt{(1-5{)}^2+(-2-6{)}^2}=\sqrt{16+64}=\sqrt{80}$

The distance between $CD=\sqrt{(5-1{)}^2+(6-8{)}^2}=\sqrt{16+4}=\sqrt{20}$

The distance between $DA=\sqrt{(1+3{)}^2+(8-0{)}^2}=\sqrt{16+64}=\sqrt{80}$

The distance between $AC=\sqrt{(-3-5{)}^2+(0-6{)}^2}=\sqrt{100}=10$

The distance between $BD=\sqrt{(1-1{)}^2+(-2-8{)}^2}=\sqrt{100}=10$

As the opposite sides $AB=CD$ and $BC=DA$ and the diagonals $AC=BD$.

Therefore, the quadrilateral which is formed by the points $A,B,C,D$ is a rectangle.

Hence, this is the answer.