#### Question

Name the quadrilateral formed, if any, by the following points, and given reasons for your answers:

A(-1,-2) B(1, 0), C (-1, 2), D(-3, 0)

#### Solution

A (-1,-2) , B(1,0), C(-1,2), D(-3,0)

Let A, B, C and D be the four vertices of the quadrilateral ABCD.

We know the distance between two points `P(x_1, y_1)` and `Q(x_2, y_2)`is given by distance formula:

`PQ = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`

Hence

`=> AB= sqrt((1 - (-1))^2 + (0 - (-2))^2)`

`=> AB= sqrt((2)^2 + (2)^2)`

`=> AB = sqrt(4+ 4)`

`=> AB = 2sqrt2`

Similarly

`=> BC= sqrt((-1)-1)^2 + (2 - 0)^2`

`=> BC= sqrt((-2)^2 + (2)^2)`

`=> BC = sqrt(4 + 4)`

`=> BC= sqrt8`

Similarly,

`=> CD = sqrt((-3)-(-1)^2 + (0 - (2))^2)`

`=> CD = sqrt((-2)^2 + (-2)^2)`

`=> CD = sqrt(4 + 4)`

`=> CD = sqrt8`

`=> CD = 2sqrt2`

Also

`=> DA = sqrt(((-1)-(-3))^2 + (0 -(-2))^2)`

`=> DA = sqrt((2)^2 + (2)^2)`

`=> DA = sqrt(4 + 4)`

`=> DA = sqrt8`

`=> DA = 2sqrt2`

Hence from above we see that all the sides of the quadrilateral are equal. Hence it is a square.