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प्रश्न
If (−1, 2), (2, −1) and (3, 1) are any three vertices of a parallelogram, then
विकल्प
a = 2, b = 0
a = −2, b = 0
a = −2, b = 6
a = 6, b = 2
None of these
उत्तर
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
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