Hefty construction or Simple formula?

Geometry Level 2

If A B = 10 , B C = 7 , C D = 8 , A D = 5 AB=10 , BC= 7, CD=8,AD=5 , then the area of the cyclic quadrilateral can be represented as A 7 A\sqrt{7} .

Find A A .

All the sides are in cm \text{cm} .
Image credit: Art of Problem Solving .


The answer is 20.

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Rishik Jain by Rishik Jain , 21, India

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1 solution

Akshat Sharda
Feb 26, 2016

By Brahmagupta's formula for area of cyclic quadrilateral ( A ) (A) ,

{ A = ( s − a ) ( s − b ) ( s − c ) ( s − d ) s = a + b + c + d 2 A = ( 15 − 5 ) ( 15 − 10 ) ( 15 − 7 ) ( 15 − 8 ) = 20 7 ⇒ 20 \begin{cases} A =\sqrt{(s-a)(s-b)(s-c)(s-d)} \\ s =\frac{a+b+c+d}{2} \end{cases} \\ \begin{aligned} A & = \sqrt{(15-5)(15-10)(15-7)(15-8)} \\ & = 20 \sqrt{7} \\ & \Rightarrow \boxed{\color{#D61F06}{20}} \end{aligned}

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