# Ex.8.2 Q5 Quadrilaterals Solution - NCERT Maths Class 9

## Question

In a parallelogram \(ABCD\), \(E\) and \(F\) are the mid-points of sides \(AB\) and \(CD\) respectively (see the given figure). Show that the line segments \(AF\) and \(EC\) trisect the diagonal \(BD\).

## Text Solution

**What is known?**

In a parallelogram \(ABCD\), \(E\) and \(F\) are the mid-points of sides \(AB\) and \(CD\) respectively**.**

**What is unknown?**

How we can show that the line segments \(AF\) and \(EC\) trisect the diagonal \(BD.\)

**Reasoning:**

In a quadrilateral if one pair of opposite sides is parallel and equal to each other. Then it is a parallelogram. Also by converse of mid-point theorem, we know that a line drawn through the mid-point of any side of a triangle and parallel to another side, bisects the third side.

**Steps:**

\(ABCD\) is a parallelogram.

\(AB\) \( \parallel \) \(CD\)

And hence, \(AE\) \( \parallel \) \(FC\)

Again, \(AB = CD\) (Opposite sides of parallelogram \(ABCD\))

\(\begin{align}\frac{1}{2}{AB}=\frac{1}{2}{CD}\end{align}\)

\(AE = FC\) (\(E\) and \(F\) are mid-points of side \(AB\) and \(CD\))

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

\(\therefore\) \(AF\) \( \parallel \) \(EC\) (Opposite sides of a parallelogram)

In \(\rm \Delta DQC,\) \(F\) is the mid-point of side \(DC\) and \(FP\) \( \parallel \) \(CQ\) (as \(AF\) \( \parallel \) \(EC\)). Therefore, by using the converse of mid-point theorem, it can be said that \(P\) is the mid-point of \(DQ\).

\(\therefore\) \(DP = PQ \)... (1)

Similarly, in \(\rm \Delta APB,\) \(E\) is the mid-point of side \(AB\) and \(EQ\) \( \parallel \) \(AP\) (as \(AF\) \( \parallel \) \(EC\)).

Therefore, by using the converse of mid-point theorem, it can be said that \(Q\) is the mid-point of \(PB\).

\(\therefore\) \(PQ = QB\) ... (2)

From Equations (1) and (2),

\(DP = PQ = BQ\)

Hence, the line segments \(AF\) and \(EC\) trisect the diagonal \(BD\).