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Question

(i) If the points A (a, −11), B (5, b), C(2, 15) and D (1, 1) are the vertices of a parallelogram ABCD, find the values of a and b.

(ii) Point A(3, 1), B(5, 1), C(a, b) and D(4, 3) are vertices of a parallelogram ABCD. Find the values of a and b.

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Solution

(i)

Let ABCD be a parallelogram in which the co-ordinates of the vertices are A (a,−11); B (5, b); C (2, 15) and D (1, 1).

Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide.

In general to find the mid-point of two pointsand we use section formula as,

The mid-point of the diagonals of the parallelogram will coincide.

So,

Therefore,

Now equate the individual terms to get the unknown value. So,

Similarly,

Therefore,

(ii) Given: Point A(3, 1), B(5, 1), C(a, b) and D(4, 3) are vertices of a parallelogram ABCD.

Diagonals of a parallelogram bisect each other.

∴ Mid point of AC = Mid point of BD

Mid point of x1, y1 and x2, y2 is x1+x22, y1+y22.Mid point of AC=3+a2, 1+b2Mid point of BD=5+42, 1+32 =92, 42=92, 23+a2, 1+b2=92, 23+a2=92 and 1+b2=23+a=9 and 1+b=4a=6 and b=3

Hence, the values of a and b is 6 and 3, respectively.

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