Question

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)

Answer 1

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.