Name the type of quadrilateralformed, if any, by the followingpoints, and give reasons foryour answer:
(i) (– 1, – 2), (1, 0), (– 1, 2), (– 3, 0)
(ii) (–3, 5), (3, 1), (0, 3), (–1, – 4)
(iii) (4, 5), (7, 6), (4, 3), (1, 2)
(i) Let the points ( - 1, - 2), (1, 0), ( - 1, 2), and ( - 3, 0) be representing the vertices A, B, C, and D of the givenquadrilateral respectively
AB = [(-1- 1)2 + (-2- 0)2]1/2
= 
= 
= 2
BC = [(1 + 1)2 + (0- 2)2]1/2
= 
= 
= 2
CD = [(-1 + 3)2 + (2- 0)2]1/2
= 
= 
= 2
AD = [(-1+ 3)2 + (-2- 0)2]1/2
= 
= 
= 2
Diagonal AC = [(-1 + 1)2 + (-2 - 2)2]1/2
= 
= 4
Diagonal BD = [(1 + 3)2 + (0- 0)2]1/2
= 
= 4
It is clear that all sides of this quadrilateral are of the same length and the diagonals are of the samelength. Therefore, the given points are the vertices of a square
(ii)Let the points (- 3, 5), (3, 1), (0, 3), and ( - 1, - 4) be representing the vertices A, B, C, and D of the given quadrilateral respectively
AB = [(-3- 3)2 + (5- 1)2]1/2
= 
= 
= 2
BC = [(3- 0)2 + (1- 3)2]1/2
= 
= 
CD = [(0 + 1)2 + (3+ 4)2]1/2
= 
= 
= 5
AD = [(-3+ 1)2 + (5+ 4)2]1/2
= 
= 
We can observe that all sides of this quadrilateral are of different lengths.
Therefore, it can be said that it is only ageneral quadrilateral, and not specific such as square, rectangle, etc
(iii)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 = [(4- 7)2 + (5 - 6)2]1/2
= 
= 
BC = [(7- 4)2 + (6 - 3)2]1/2
= 
= 
CD = [(4- 1)2 + (3 - 2)2]1/2
= 
= 
AD = [(4- 1)2 + (5 - 2)2]1/2
= 
= 
Diagonal AC = [(4 - 4)2 + (5- 3)2]1/2
= 
= 2
Diagonal CD = [(7 - 1)2 + (6 - 2)2]1/2
= 
= 
= 13
We can observe that opposite sides of this quadrilateral are of the same length.
However, the diagonals are ofdifferent lengths. Therefore, the given points are the vertices of a parallelogram