Question

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

(5, 2), (1, 5), (-2, 1) and (2, -2)

Answer 1

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

The distance between $AB=\sqrt{(5-1{)}^2+(2-5{)}^2}=\sqrt{4^2+(-3{)}^2}=\sqrt{16+9}=\sqrt{25}=5$

The distance between $BC=\sqrt{(1+2{)}^2+(5-1{)}^2}=\sqrt{9+16}=5$

The distance between $CD=\sqrt{(-2--2{)}^2+(1+2{)}^2}=\sqrt{16+9}=5$

The distance between $DA=\sqrt{(2-5{)}^2+(-2-2{)}^2}=\sqrt{9+16}=5$

The distance between $AC=\sqrt{(5+2{)}^2+(2-1{)}^2}=\sqrt{49+1}=\sqrt{50}$

The distance between $BD=\sqrt{(1-2{)}^2+(5+2{)}^2}=\sqrt{1+49}=\sqrt{50}$

As the sides are equal $AB=BC=CD=DA$

The diagonals are also equal $AC=BD$.

Therefore, the given points represents square.

Hence, this is the answer.