Show that points P(2,-2) , Q (7,3) ,R(11,-1) and S(6,-6)
are vertices of a parallelogram.

Explanation

P(2,-2),Q(7,3), R(11,-1) and S(6,-6)
By distance formula ,
` PQ = sqrt((7-2)^(2) + [3-(-2)]^(2))`
` therefore = sqrt(5^(2) + 5^(2))`
` therefore PQ = sqrt(25 + 25 )`
`therefore PQ = sqrt(50)`
` therefore PQ= sqrt(5xx5xx2)`
` therefore PQ 5sqrt(2)`...(1)
`QR = sqrt((11-7)^(2) + (-1-3)^(2)) `
` therefore QR = sqrt(4^(2) + (-4)^(2))`
` therefore QR = sqrt(16+16)`
` therefore QR = sqrt(32)`
` therefore QR = sqrt(2xx2xx2xx2xx2xx)`
` therefore QR = 4sqrt(2)` ...(2)
` RS = sqrt(6-11)^(2) + [-6-(-1)]^(2))`
` therefore RS = sqrt(-5)^(2) + (-5)^(2))`
` therefore RS = sqrt(25 + 25)`
` therefore RS= sqrt(50)`
` therefore RS = sqrt(5xx5xx2)`
` therefore RS = 5sqrt(2)` ...(3)
`PS = sqrt((6-2)^(2) + [-6-(-2)]^(2))`
`therefore PS = sqrt(4^(2) + (-4)^(2))`
` therefore sqrt(16 + 16) `
` therefore PS = sqrt(32)`
` therefore PS = sqrt(2xx2xx2xx2xx2)`
` therefore PS 2xx2xxsqrt(2)`
` therefore PS = 4 sqrt(2)` ....(4)
In `square ` PQRS
` PQ = RS " " ` ...[From (1) and (3)]
`QR = PS " "` ...[From (2) and (4)]
A quadrilateral is a parallelogram , if both the pairs of its opposite
sides are congruent
` therefore square` PQRS is parallelogram.
` therefore P , Q , R ` and S are vertices of a parallelogram .