# Ex.6.4 Q5 The-Triangle-and-its-Properties Solutions - NCERT Maths Class 7

## Question

\(ABCD\) is quadrilateral. Is \(AB \!+\! BC \!+\! CD \!+\! DA\! <\! 2 (AC \!+\! BD)?\)

## Text Solution

**What is known? **

\(ABCD\) is a quadrilateral. \(DB\) and \(AC\) are diagonals.

**What is unknown? **

Is \(AB \!+\! BC \!+\! CD \!+\! DA \!<\! 2 (AC \!+\! BD)?\)

**Reasoning:**

In this question, it is asked to check Is \(AB \!+\! BC \!+\! CD \!+\! DA \!<\!2(AC \!+\! BD)\) or not.

This question is based on the property that the sum of lengths of two sides of a triangle is always greater than the third side.\(“O”\) is the centre of the quadrilateral. Now visually identify that the quadrilateral \(ABCD\) is divided by diagonals \(AC\) and \(BD\) into four triangles. Now, take each triangle separately i.e. triangle \(AOB, COD, BOC \) and \(AOD\) and apply the above property and then add \(L.H.S\) and \(R.H.S\) of the equation formed.

**Steps:**

In triangle \(AOB,\)

\(AB < OA + OB{\rm{ }} \qquad \ldots \ldots .{\rm{ }}\left( 1 \right)\)

In triangle \(COD,\)

\(CD < OC + OD{\rm{ }} \qquad \ldots \ldots .{\rm{ }}\left( 2 \right)\)

In triangle \(AOD,\)

\(DA < OD + OA{\rm{ }} \qquad \ldots \ldots .{\rm{ }}\left( 3 \right)\)

In triangle \(COB,\)

\(BC < OC + OB{\rm{ }} \qquad \ldots \ldots .{\rm{ }}\left( 4 \right)\)

Adding equation \((1),\) \((2),\) \((3)\) and \((4)\) we get,

\[\begin{align}& \left[ \begin{array}& AB\!+\!BC\!+\!\\CD\!+\!DA \end{array} \right]\!<\!\left[ \begin{array}& OA\!+\!OB\!+\! \\ OC\!+\!OD\!+\!\\OD\!+\!OA\!+\! \\ OC\!+\!OB \\ \end{array} \right] \\ =& \left[ \begin{array}& AB\!+\!BC\!+\!\\CD\!+\!DA\end{array} \right]<\left[ \begin{array}&2OA\!+\!2OB\!+\!\\2OC\!+\!2OD\end{array} \right] \\ =& \left[ \begin{array} & AB\!+\!BC\!+\! \\ CD\!+\!DA \\ \end{array} \right]<\left[ \begin{array}&2(OA\!+\!OB)\!+\!\\2(OC\!+\!OD)\end{array} \right] \\ = & AB\!+\!BC\!+\!CD\!+\!DA\!<\!2(AC\!+\!BD) \\ \end{align}\]

Yes, it is true.

Useful Tip

Whenever you encounter problems of this kind, it is best to think of the property based on sum of lengths of any two sides of a triangle is always greater than the third side.