Ex.6.4 Q4 The-Triangle-and-its-Properties Solutions NCERT Maths Class 7

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Question

\(\rm{}ABCD\) is a quadrilateral. Is \(\rm{}AB + BC + CD + DA > AC + BD?\)

Text Solution

What is known?

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

What is unknown?

Is \(AB + BC + CD + DA > AC + BD?\)

Reasoning:

In this question, it is asked to check Is \(\rm{}AB + BC + CD + DA > AC + BD\)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. Now visually identify that the quadrilateral \(\rm{}ABCD\) is divided by diagonals \(\rm{}AC\) and \(\rm{}BD\) into four triangles. Now, take each triangle separately and apply the above property and then add \(\rm{}L.H.S\) and \(\rm{}R.H.S\) of the equation formed.

Steps:

In triangle \(\rm{}ABC,\)

          \(AB + BC > AC{\rm{ }} \qquad \ldots \ldots \ldots \ldots \ldots \ldots .{\rm{ }}\left( 1 \right)\)

In triangle \(\rm{}ADC,\)

          \(AD + CD > AC{\rm{ }} \qquad \ldots \ldots \ldots \ldots \ldots \ldots .{\rm{ }}\left( 2 \right)\)

In triangle \(\rm{}ADB,\)

          \(AD + AB > DB{\rm{ }} \qquad \ldots \ldots \ldots \ldots \ldots \ldots .{\rm{ }}\left( 3 \right)\)

In triangle \(\rm{}DCB,\)

          \(DC + CB > DB{\rm{ }} \qquad \ldots \ldots \ldots \ldots \ldots \ldots .{\rm{ }}\left( 4 \right)\)

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

\[\begin{align}&\rm{}AB + BC + AD + CD + AD + AB + DC + CB > AC + AC + DB + DB\\&\rm{}AB + AB + BC + BC + CD + CD + AD + AD > 2AC + 2DB\\&\rm{}2AB + 2BC + 2CD + 2AD > 2AD + 2DB\\&\rm{}AB + BC + CD + AD > AC + DB\end{align}\]

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.

  
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