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Question

If (1,2),4,y, x,6 and 3,5 are the vertices of a parallelogram taken in order, find x and y.


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Solution

Step 1: Find the midpoint of diagonals of parallelogram

Let A=(1,2), B=4,y,C=x,6, D=3,5

AC and BD are the diagonals of the parallelogram

Let M and N be the midpoints of AC and BD respectively

By Midpoint formula

M=xA+xC2,yA+yC2

M=1+x2,2+62

M=1+x2,4...(i)

N=xB+xD2,yB+yD2

N=4+32,y+52

N=72,y+52...(ii)

Step 2: Use property of parallelogram to find value of x and y

The diagonals of a parallelogram bisect each other.

Hence, the midpoints of the diagonals are coincident.

M and N are equal say point O

Hence, the co-ordinates of M and N are equal

xM=xN and yM=yN

1+x2=72 …[From(i)]

1+x=7

x=6

y+52=4 …[From(ii)]

y+5=8

y=3

Hence, the value of x is 6 and y is 3


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