# Ex.8.2 Q6 Quadrilaterals Solution - NCERT Maths Class 9

## Question

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

## Text Solution

**What is known?**

\(ABCD\) is a quadrilateral in which \(P, Q, R,\) and \(S\) are the mid-points of sides \(AB, BC, CD,\) and \(DA\) respectively.

**What is unknown?**

How we can show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

**Reasoning:**

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. In a quadrilateral if one pair of opposite sides is parallel and equal to each other then it is a parallelogram.

**Steps:**

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 \(\Delta {ABD,}\) \(S\) and \(P\) are the mid-points of \(AD\) and \(AB\) respectively. Therefore, by using mid-point theorem, it can be said that

\[ \begin{align} SP \parallel BD \; and\; SP\!=\!\frac{1}{2} BD \quad\!\!... (1) \end{align}\]

Similarly, in \(\Delta \) \(BCD\),

\[\begin{align} QR \parallel BD \; and\; QR \!=\!\frac{1}{2} BD \quad\!\!... (2) \end{align}\]

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

\(SP\) \(\parallel \)\(QR\) and \(SP = QR\)

In quadrilateral \(SPQR\), one pair of opposite sides is equal and parallel to each other. Therefore, \(SPQR\) is a parallelogram.

We know that diagonals of a parallelogram bisect each other.

Hence, \(PR\) and \(QS\) bisect each other.