Consider a parallelogram whose vertices are A (1, 2), B (4, y), C (x, 6) and D (3, 5) taken in order

What is the point of intersection of the diagonals?

Concept:

• A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length and the opposite angles are equal in measure.
• Diagonals of parallelogram bisect each other.
• Midpoint formula:

Suppose x, y are midpoints then) 

Where,

 = coordinates of first point

 = coordinates of second point

Calculation:

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 = 

Midpoint of BD =           (Using midpoint formula)

We can see that,

Midpoint of AC = midpoint of BD

                  (Equating respective co-ordinates)

The point of intersection of diagonals = Midpoint of AC

(m, n) = 

= 

= 

Hence, option 1 is correct.

Additional point:

• Suppose p & q are length of diagonals and a & b are length of sides of parallelogram then we have relation