**Q) ABCD is a rectangle formed by the points A (−1, −1), B (−1, 6), C (3, 6) and D (3, −1). P, Q, R and S are mid-points of sides AB, BC, CD and DA respectively. Show that diagonals of the quadrilateral PQRS bisect each other.**

**Ans: **Let’s make a diagram for the given question:

Let’s start fiding coordinates of points P, Q, R and S.

We know that the coordinates of a midpoint lying between (x1, y1) and (x2, y2) is given by:

M(x,y) =

Since P is the midpoint of AB, ∴ P = = (- 1, )

Similarly, Q is the midpoint of BC, ∴ P = = (1, 6)

Similarly, R is the midpoint of CD, ∴ P = = (3, )

Similarly, S is the midpoint of DA, ∴ P = = (1, – 1)

If Diagonals of PQRS bisect each other, O will be the midpoint of PR and QS both.

Let’s check if this relationship is verified:

If O is the midpoint of PR, then its coordinates are given by: = (1,)

If O is the midpoint of QS, then its coordinates are given by: = (1,)

Since both points coordinates match with each other, it confirms that O is midpoint of PR & QS

**Therefore, diagonals of quadrilateral PQRS bisect each other.**

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