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 parallelogram
• C. a square
• D. a rhombus which is not a square.

Answer 1

The correct option is C a square

Let $A(2,1),B(1,4),C(4,5)$ and $D(5,2)$ be the four coordinates then using

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

$AB=\sqrt{{(1-2)}^2+{(4-1)}^2}=\sqrt{1+9}=\sqrt{10}$

$BC=\sqrt{{(4-1)}^2+{(5-4)}^2}=\sqrt{9+1}=\sqrt{10}$

$CD=\sqrt{{(5-4)}^2+{(2-5)}^2}=\sqrt{1+9}=\sqrt{10}$

$DA=\sqrt{{(5-2)}^2+{(2-1)}^2}=\sqrt{9+1}=\sqrt{10}$

$\therefore AB=BC=CD=DA$

Hence $ABCD$ form a square.