UPSC NDA Math Model Paper 7

Here, A (1, 2), B (4, y), C (x, 6) and D (3, 5) are the vertices of parallelogram.
AB = CD and AD = BC (sides of parallelogram)
AC and BD are the diagonals which bisect each other.
Suppose diagonals intersect each other at (m, n)
Midpoint of AC=(

1+x

2

,

6+2

2

)
Midpoint of BD=(

4+3

2

,

y+5

2

) (Using midpoint formula)
We can see that,
Midpoint of AC = midpoint of BD
(

1+x

2

,

6+2

2

)=(

4+3

2

,

y+5

2

)
∴

1+x

2

=

7

2

‌and‌

8

2

=

y+5

2

(Equating respective co-ordinates)
⇒1+x=7‌and‌8=y+5
⇒x=6‌and‌y=3 The point of intersection of diagonals = Midpoint of AC
(m,n)=(

1+x

2

,

6+2

2

)
=(

1+6

2

,

6+2

2

)
=(

7

2

,4)
Hence, option 1 is correct.