Question

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

(12, 9), (20, -6), (5, -14) and (-3, 1)

Answer 1

Let us consider the given points as $A(12,9);B(20,-6);C(5,-14);D(-3,1)$.

The distance between $AB=\sqrt{(12-20{)}^2+(9+6{)}^2}=\sqrt{64+225}=\sqrt{289}=17$

The distance between $BC=\sqrt{(20-5{)}^2+(-6+14{)}^2}=\sqrt{225+64}=\sqrt{289}=17$

The distance between $CD=\sqrt{(5+3{)}^2+(-14-1{)}^2}=\sqrt{64+225}=\sqrt{289}=17$

The distance between $DA=\sqrt{(-3-12{)}^2+(1-9{)}^2}=\sqrt{225+64}=\sqrt{289}=17$

The distance between $AC=\sqrt{(12-5{)}^2+(9+14{)}^2}=\sqrt{7^2+{23}^2}=\sqrt{49+529}=\sqrt{578}$

The distance between $BD=\sqrt{(20+3{)}^2+(-6-1{)}^2}=\sqrt{{23}^2+(-7{)}^2}=\sqrt{578}$

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.