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Question

If A (–2, –1), B (a, 0), C (4, b) and D (1, 2) are the vertices of a parallelogram, find the values of a and b.

[5]

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Solution

We know that the diagonals of a parallelogram bisect each other. Therefore, the coordinates of the mid-point of AC are same as the coordinates of the mid-point of BD.

The coordinates of the mid-point of a line formed by joining two points (x1,y1) and (x2,y2) are (x1+x22,y1+y22)

[1]


Midpoint of AC = (2+42,1+b2)

Midpoint of BD = (a+12,0+22)
[1]

(2+42,1+b2) = (a+12,0+22)

(1,b12)=(a+12,1)

a+12 = 1 and b12 = 1

⇒ a + 1 = 2 and b - 1 = 2

⇒ a = 1 and b = 3
[1]

We know that the diagonals of a parallelogram bisect each other. Therefore, the coordinates of the mid-point of AC are same as the coordinates of the mid-point of BD.

The coordinates of the mid-point of a line formed by joining two points (x1,y1) and (x2,y2) are (x1+x22,y1+y22)

Midpoint of AC = (2+42,1+b2)

Midpoint of BD = (a+12,0+22)

(2+42,1+b2) = (a+12,0+22)
[1]

(1,b12)=(a+12,1)

a+12 = 1 and b12 = 1

⇒ a + 1 = 2 and b - 1 = 2

⇒ a = 1 and b = 3
[1]


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