# Ex.9.4 Q3 Areas of Parallelograms and Triangles Solution - NCERT Maths Class 9

## Question

In the following figure, \(ABCD\), \(DCFE\) and \(ABFE\) are parallelograms. Show that ar (\( \Delta \rm{ADE} \)) \(=\) ar (\( \Delta \rm{BCF} \)).

## Text Solution

**What is known?**

\(ABCD\),\( DCFE\) and \(ABFE\) are parallelograms.

**What is unknown?**

How we can show that \(\text{ar }\left( \Delta \text{ADE} \right)\text{ = ar }\left( \Delta {BCF} \right)\text{.}\)

**Reasoning:**

We can see that sides of triangles \(ADE\) and \(BCF\) are also the opposite sides of the given parallelogram. Now we can show both the triangles congruent by SSS congruency. We know that congruent triangles have equal areas.

**Steps:**

It is given that \(ABCD\) is a parallelogram. We know that opposite sides of a parallelogram are equal.

\( \therefore \rm A D=B C \ldots(1) \)

Similarly, for parallelograms \(DCFE\) and \(ABFE\), it can be proved that

\(DE=CF\).\(...(2)\)

And, \( EA = FB\)\( ... (3)\)

In \( \Delta {ADE} \) and \( \Delta {BCF} \),

\(AD = BC\) [Using equation (\(1\))]

\(DE = CF\) [Using equation (\(2\))]

\(EA = FB\) [Using equation (\(3\))]

\( \therefore \Delta ADE \cong \Delta BCF\!\! \begin{pmatrix} \!\text { SSS} \!\\ \!\text{congruence } \!\\ \!\text{rule } \! \end{pmatrix} \)

The SSS Rule States that : If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.

\[\therefore \text { Area }(\Delta {ADE})=\text { Area }(\Delta {BCF}) \]