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Question

Using the section formula, show that the points
A(1, 0), B(5, 3), C(2, 7) and D(-2, 4) are vertices of a parallelogram taken in order.

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Solution

Diagonals of a paralleogram bisect each other.
So, the midpoint of the diagonal AC must coincide with the midpoint of diagonal BD.
Using section formula we can write,

(lx2+mx1)/(l+m),(ly2+my1)/(l+m)

l = 1 and m = 1

The midpoint of the diagonal AC divides the line AC in the ratio 1:1

= 1(2)+1(1)(+1 , 1(7)+1(0)1+1

= 2+12 , 7+02

= 32, 72 ------- (1)

The midpoint of the diagonal BD divides the line BD in the ratio 1:1

= 1(2)+1(5)1+1 , 1(4)+1(3)1+1

= 2+52 , 4+32

= 32, 72 ------- (2)

So, the diagonals bisect each other at a common point.
Hence, the given vertices form a parallelogram.

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