# 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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