Question

Find the area of a quadrilateral ABCD whose sides AB = 13 cm, BC = 12 cm, CD = 9 cm, DA = 14 cm and diagonal BD = 15 cm.

Solution

For ∴ABD
a = 13 cm b = 14 cm c = 15 cm

$therefore space space space space straight s equals fraction numerator straight a plus straight b plus straight c over denominator 2 end fraction equals fraction numerator 13 plus 14 plus 15 over denominator 2 end fraction equals 21 space cm therefore space space space space Area space equals space square root of straight s left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis end root space space space space space space space space space space space space equals square root of 21 left parenthesis 21 minus 13 right parenthesis left parenthesis 21 minus 14 right parenthesis left parenthesis 21 minus 15 right parenthesis end root space space space space space space space space space space space space equals space square root of 21 cross times 8 cross times 7 cross times 6 end root space space space space space space space space space space space space equals space 84 space cm squared$
For ΔBCD
a = 9 cm b = 12 cm c = 15 cm
$therefore space space space space space straight s equals fraction numerator straight a plus straight b plus straight c over denominator 2 end fraction equals fraction numerator straight a plus 12 plus 15 over denominator 2 end fraction equals 18 space cm therefore space space space space Area space equals space square root of straight s left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis end root space space space space space space space space space space space equals square root of 18 left parenthesis 18 minus 9 right parenthesis left parenthesis 18 minus 12 right parenthesis left parenthesis 18 minus 15 right parenthesis end root space space space space space space space space space space equals square root of 18 cross times 9 cross times 6 cross times 3 end root space equals space 54 space cm squared$

Now, area of quadrilateral ABCD
= area of ΔABD + area of ΔBCD = 84 cm² + 54 cm²= 138 cm²

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Statistics

Find the area of a quadrilateral ABCD whose sides AB = 13 cm, BC = 12 cm, CD = 9 cm, DA = 14 cm and diagonal BD = 15 cm.

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