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RD Chapter 12 Heron s Formula Ex 12.6 Solutions

Question - 11 : - Prove that in a quadrilateral the sum of all the sides is greater than the sum of its diagonals.

Answer - 11 : -

Given : In quadrilateral ABCD, AC and BD are its diagonals,
 
To prove : AB + BC + CD + DA > AC + BD
Proof: In ∆ABC,
AB + BC > AC …(i)
(Sum of any two sides of a triangle is greater than its third side)
Similarly, in ∆ADC,
DA + CD > AC …(ii)
In ∆ABD,
AB + DA > BD …(iii)
In ∆BCD,
BC + CD > BD …(iv)
Adding (i), (ii), (iii) and (iv)
2(AB + BC + CD + DA) > 2AC + 2BD
⇒ 2(AB + BC + CD + DA) > 2(AC + BD)
∴ AB + BC + CD + DA > AC + BD

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