# 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.

**☛ Check: **NCERT Solutions for Class 9 Maths Chapter 8

**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.

**☛ Related Questions:**

- ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.
- ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.
- ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F (see Fig. 8.30). Show that F is the mid-point of BC
- ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show thati) D is the mid-point of ACii) MD ⊥ ACiii) CM = MA = 1/2 AB