Name the type of quadrilateral formed, if any, by the following point, and give reasons for your answer (4, 5), (7, 6), (4, 3), (1, 2)

#### Solution

Let the points (4, 5), (7, 6), (4, 3), and (1, 2) be representing the vertices A, B, C, and D of the given quadrilateral respectively.

`:.AB = sqrt((4-7)^2+(5-6)^2) = sqrt((-3)^2+(-1)^2) = sqrt(9+1) = sqrt10`

`BC =sqrt((7-4)^2+(6-3)^2) = sqrt((3)^2+(3)^2) = sqrt(9+9) = sqrt18`

`CD = sqrt((4-1)^2+(3-2)^2) = sqrt((3)^2+(1)^2) = sqrt(9+1) = sqrt10`

`AD = sqrt((4-1)^2+(5-2)^2) = sqrt((3)^2+(3)^2) = sqrt(9+9) = sqrt18`

Diagonal AC =`sqrt((4-4)^2+(5-3)^2) = sqrt((0)^2+(2)^2) = sqrt(0+4) = 2`

Diagonal CD =`sqrt((7-1)^2 + (6-2)^2) = sqrt((6)^2+(4)^2) = sqrt(36+16) = sqrt52 = 13sqrt2`

It can be observed that opposite sides of this quadrilateral are of the same length. However, the diagonals are of different lengths. Therefore, the given points are the vertices of a parallelogram.