Question

Show that the following points are the vertices of a rectangle:

A(2, –2), B(14, 10), C(11, 13) and D(–1,1)

Answer 1

The given points are  A (2, -2), B(14,10), C(11,13) and D(-1,1).

`AB = sqrt((14-2)^2 +{10-(-2)}^2) = sqrt((12)^2 +(12)^2) =sqrt(144+144) = sqrt(288) =12 sqrt(2)  units`

`BC = sqrt(( 11-14)^2 +(13-10)^2 ) = sqrt((-3)^2 +(3)^2) = sqrt(9+9) = sqrt(18) = 3 sqrt(2)   units`

` CD = sqrt((-1-11)^2 +(1-13)^2) = sqrt((-12)^2 +(-12)^2) = sqrt(144+144) = sqrt(288) = 12 sqrt(2)  units`

`AD = sqrt((-1-2)^2 +{1-(-2)}^2) = sqrt((-3)^2 +(3)^2) = sqrt(9+9) = sqrt(18) =3 sqrt(2)  units`

`Thus  AB =CD = 12 sqrt(2)    "units and " BC =AD = 3 sqrt(2) units`

Also , 

`AC = sqrt((11-2)^2 +{ 13-(-2)}^2) = sqrt((9)^2 +(15)^2) = sqrt(81+225) = sqrt(306) = 3 sqrt(34)  units `

` BD = sqrt((-1-14)^2 +(1-10)^2) = sqrt((-15)^2 +(-9)^2) = sqrt(81+225) = sqrt(306) =3 sqrt(34)  units`

Also, diagonal AC = diagonal BD

Hence, the given points from a rectangle