ABCD is a quadrilateral in which …

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ABCD is a quadrilateral in which AC & BD are two diagonals .Show that AB+BC+CD+AD>AC+BD

Posted by Shivraj Patil 5 years, 7 months ago

CBSE > Class 09 > Mathematics

1 answers

Sia ? 5 years, 7 months ago

Given: A quadrilateral ABCD.
To prove: AB + BC + CD + DA > AC + BD
Proof: In $\Delta ABC$ , we have

AB + BC > AC…(1) [ $\because$ Sum of the lengths of any two sides of a triangle must be greater than the third side]
In $\Delta BCD$ , we have
BC + CD > BD...(2) [Same reason]
In $\Delta CDA$ , we have
CD + DA > AC…(3) [Same reason]
In $\Delta DAB$ , we have
AD + AB > BD…(4) [Same reason]
Adding (1), (2), (3) and (4), we get
AB + BC + BC + CD + CD + DA + AD + AB > AC + BD + AC + BD
$\Rightarrow$ 2AB + 2BC + 2CD + 2DA > 2AC + 2BD
$\Rightarrow$ 2(AB + BC + CD + DA) > 2(AC + BD)
$\Rightarrow$ AB + BC + CD + DA > AC + BD
Hence, proved.

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CBSE > Class 09 > Mathematics