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Q.

If (3, -4) and (-6, 5) are the extremities of the diagonal of a parallelogram and (-2, 1) is its third vertex, then its fourth vertex is:

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a

(-1, 0)

b

(0, -1)

c

None of these

d

(-1 , 1)

answer is A.

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Detailed Solution

Given that, (3, -4) and (-6, 5) are the extremities of the diagonal and (-2, 1) is its third vertex.
We need to find the 4th vertex.
Let A(3, -4) and C(-6, 2) be the extremities of diagonal AC and B(-1,-3), D(x,y) the extremities of diagonal BD.
Midpoint of two point

{(x}_{1}, y_{1})

&

{(x}_{2}, y_{2})

is ,

(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})

.
Since the diagonals of a parallelogram bisect each other.
Therefore,
Coordinates of midpoint of AC = Coordinates of midpoint of BD.

(\frac{3 + (- 6)}{2}, \frac{(- 4) + 2}{2}) = (\frac{(- 1) + x}{2}, \frac{(- 3) + y}{2})

Equating x coordinates,

\Rightarrow \frac{3 - 6}{2} = \frac{x - 1}{2}

\Rightarrow \frac{- 3}{2} = \frac{x - 1}{2}

\Rightarrow x = - 2

And, equating y coordinates,

\Rightarrow \frac{- 4 + 2}{2} = \frac{y - 3}{2}

\Rightarrow \frac{- 2}{2} = \frac{y - 3}{2}

\Rightarrow y = 1

Thus, the coordinates of the fourth point is (-2,1).
Therefore, option (1) is correct.

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