If `P(2,\ -1),\ \ Q(3,\ 4),\ \ R(-2,\ 3)` and `S(-3,\ -2)` be four points in a plane, show that `P Q R S` is a rhombus but not a square. Find the area of the rhombus.

Explanation

Let A(2, -1), B(3, 4), C(-2, 3) and D(-3, -2) be the angular points of a quad. ABCD Join AC and BD. Then,
`AB = sqrt((3-2)^(2) + (4+1)^(2))`
` = sqrt(1^(2) + 5^(2)) = sqrt(26)` units,
`BC = sqrt((-2-3)^(2) + (3-4)^(2)) = sqrt((-5)^(2) +(-1)^(2)) = sqrt(26)` units,
`CD = sqrt((-3+2)^(2) + (-2-3)^(2)) = sqrt((-1)^(2) + (-5)^(2)) = sqrt(26)` units,
`DA = sqrt((-3-2)^(2) + (-2+1)^(2))= sqrt((-5)^(2) + (-1)^(2))` = sqrt(26)` units.
`therefore AB = BC =CD = DA = sqrt(26)` units.
Now, `AC = sqrt((-2-2)^(2) + (3+1)^(2))= sqrt((-4)^(2)+4^(2))`
`=sqrt(32) = 4sqrt(2)` units
and `BD = sqrt((-3-3)^(2) + (-2-4)^(2)) = sqrt((-6)^(2) + (-6)^(2) + (-6)^(2))`
`=sqrt(72) =6sqrt(2)` units.
`therefore "diagonal"AC ne` diagonal BD.
Thus, ABCD is a quadrilateral in which all sides are equal but diagonals are not equal.
`therefore` ABCD is a rhombus but not a square.
`therefore ` area of the rhombus ABCD ` = (1)/(2) xx ("product of diagonals")`
`=((1)/(2) xx AC xx BD)`
` =((1)/(2) xx 4sqrt(2) xx 6sqrt(2))` sq units
= 24 sq units.
Hence, the area of rhombus ABCD is 24 sq units.