Question

The four points whose co-ordinates are $(2,1),(1,4),(4,5),(5,2)$ form

• A. a rectangle which is not a square
• B. a trapezium which is not a paralleloglam
• C. a square
• D. a rhombus which is not a square

Answer 1

Answer: (C)

a square

$A(2,1),B(1,4),C(4,5),D(5,2)$

$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

Vertices

$AB=\sqrt{(1-2{)}^2+(4-1{)}^2}=\sqrt{10}$

$BC=\sqrt{(4-1{)}^2+(5-4{)}^2}=\sqrt{10}$

$CD=\sqrt{(5-4{)}^2+(2-5{)}^2}=\sqrt{10}$

$AD=\sqrt{(5-2{)}^2+(2-1{)}^2}=\sqrt{10}$

Diagonals

$AC=\sqrt{(4-2{)}^2+(5-1{)}^2}=\sqrt{20}$

$BD=\sqrt{(5-1{)}^2+(2-4{)}^2}=\sqrt{20}$

Since lenghts of sides are same and lengths of diagonals are same, they form a square.

Even, $(diagonal{)}^2=(side1{)}^2+(side2{)}^2\Rightarrow (\sqrt{20}{)}^2=(\sqrt{10}{)}^2+(\sqrt{10}{)}^2$