# Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

**Solution:**

In a triangle, the line segment joining the mid-points of any two sides of the triangle is parallel to the third side and is half of it.

Let ABCD is a quadrilateral in which P, Q, R, and S are the mid-points of sides AB, BC, CD, and DA respectively. Join PQ, QR, RS, SP, and BD.

In ΔABD, S and P are the mid-points of AD and AB respectively.

Therefore, by using the mid-point theorem, it can be said that

SP || BD and SP = 1/2 BD ---------- (1)

Similarly, in ΔBCD,

QR || BD and QR = 1/2 BD ---------- (2)

From equations (1) and (2), we obtain

SP || QR and SP = QR

In quadrilateral SPQR, one pair of opposite sides is equal and parallel to each other. Thus, SPQR is a parallelogram.

Since we know that diagonals of a parallelogram bisect each other we can conclude that PR and QS bisect each other as shown in the above figure.

Thus, we see that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

**Video Solution:**

## Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

### NCERT Maths Solutions Class 9 - Chapter 8 Exercise 8.2 Question 6:

**Summary:**

The line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.