Among all cyclic quadrilaterals inscribed in a circle of radius R with one of its angles equal to 120°, consider the one with maximum possible area. Its area is:
(a) √2R2 (b) 2R2 (c) √3R2 (d) 2√3R2
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1 Answers
1357Lets consider two triangles ABC and ACD.
Angle B=120 and D=60
Let side AB=a, BC=b, AD=c, CD=d, AC=t
Δ=(1/2)absin(120)+(1/2)cdsin(60)=√34(ab+cd)
we also have Δ=abt/4R + cdt/4R=t/4R(ab+cd)
so we get t=√3R
Again lets take the base as t and heights as h1 and h2
So Δ=(1/2)th1+(1/2)th2=(1/2)t(h1+h2)
Now area will be maximum when heights are maximum which gives h1+h2=2R
So Δ=√3R2
Soumyadeep Basu Thank you, Sir.Upvote·0· Reply ·2013-08-31 00:21:29