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Question
The points (3, -4) and (-6, 2) are the extremities of a diagonal of a parallelogram. If the third vertex is (-1, -3). Find the coordinates of the fourth vertex.
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Solution
Let ABCD be a parallelogram in which the coordinates of the vertices are A (-2,-1); B (1, 0) and C (4, 3). We have to find the co-ordinates of the forth vertex.
Let the fourth vertex be `D(x,y)`
Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide.
Now to find the mid-point P(x,y) of two points `A(x_1, y_1)` and `B(x_2, y_2)` we use section formula as,
`P(x,y) = ((x_1+x_2)/2, (y_1+y_2)/2)`
The mid-point of the diagonals of the parallelogram will coincide.
So,
Co-ordinate of mid-point AC = Coordinate of mid-point of BD
Therefore,
`((x+1)/2, y/2) = ((4-2)/2, (3 -1)/2)`
`((x + 1)/2,y/2) = (1,1)`
Now equate the individual terms to get the unknown value. So,
x = 1
y = 2
So the forth vertex is D(1, 2)
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