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

A(4, 5) B(7, 6), C (4, 3), D(1, 2)

#### Solution

A (4, 5), B (7,6), C(4,3), D(1,2)

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((7 - 4)^2 + (6 - 5)^2)`

`=> AB = sqrt((-3)^2 + (-3)^2)`

`=> AB = sqrt(9 + 1)`

`=> AB = sqrt(10)`

Similarly,

`=> BC = sqrt((4 - 7)^2 + (3 - 6)^2)`

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

`=> BC= sqrt(9 + 9)`

`=> BC = sqrt18`

Similarly

`=> CD = sqrt((1 - 4)^2 + (2 - 3)^2)`

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

`=> CD = sqrt(9 + 1)`

`=> CD = sqrt(9 + 1)`

`=> CD = sqrt10`

Also

`=> DA = sqrt((1 - 4)^2 + (2 -5)^2)`

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

`=> DA = sqrt(9 + 9)`

`=> DA = sqrt18`

Hence from above we see that

AB = CD and BC = DA

Hence from above we see that

AB = CD and BC = DA

Here opposite sides of the quadrilateral is equal. Hence it is a parallelogram