Q2 of 23 Page 220

The coordinates of the vertices of a quadrilateral are (2, 1), (5, 3), (8, 7), (4, 9) in order.

i) Find the coordinates of the midpoints of all sides.


ii) Prove that the quadrilateral with these midpoints as vertices is a parallelogram.

Let the mid points be A, B, C, D


Using the section formula for mid points


A(x, y)


A(x, y)


A(x, y) = (3.5,2)


B(x, y)


B(x, y)


B(x, y) = (6.5,5)


C(x, y)


C(x, y)


C(x, y) = (6,8)


D(x, y)


D(x, y)


D(x, y) = (3,5)


(ii) Length AB = √((6.5 - 3.5)2 + (5 - 2)2)


Length AB = 3√2 units


Length BC = √((6.5 - 6)2 + (5 - 8)2)


Length BC = √9.25 units


Length CD = √((6 - 3)2 + (8 - 5)2)


Length CD = 3√2 units


Length DA = √((3.5 - 3)2 + (2 - 5)2)


Length DA = √9.25 units


Since the lengths of opposite sides are equal hence it forms a parallelogram.


Hence the quadrilateral forms a parallelogram by joining the mid points .


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i) Calculate the coordinates of the point P on AB such that AP : PB = 2 : 3


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i) Find the coordinates of the fourth vertex of the small quadrilateral.


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