Question

If (−1, 2), (2, −1) and (3, 1) are any three vertices of a parallelogram, then

Options

• a = 2, b = 0

• a = −2, b = 0

• a = −2, b = 6

• a = 6, b = 2

• None of these

Answer 1

Let ABCD be a parallelogram in which the co-ordinates of the vertices are A (−1, 2);

B (2,−1) and C (3, 1). We have to find the co-ordinates of the fourth vertex.

Let the fourth vertex be D (a, b) 

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(x₁ , y₁)  and  B (x₂ , y₂ ) 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 of AC = Co-ordinate of mid-point of BD 

Therefore,

`((3-1)/2, (2+1)/2 )= ( ("a" +2 ) /2, ("b" - 1 )/2)`

`(("a" + 2 )/2,("b" - 1)/2) = (1,3/2)`

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

`("a" + 2 )/2 = 1`

           a = 0

Similarly,

`("b"-1)/2 = 3/2`

        b  =  4

So the fourth vertex is  D (0, 4) 

3rd case: C and B are opposite corners of a diagonal.

Now, mid-point of CB is `(5/2, 0)`

Mid-point of AD is `(("a"-1)/2,("b"+2)/2)`

⇒ `("a"-1)/2=5/2,("b"+2)/2=0`

⇒ a = 6, b = -2