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# help

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A rhombus is inscribed in a circle with radius r and center O such that one of the vertices lies on O and rest lie on the circumference of the circle.

Find the area of the rhombus in terms of r.

Dec 27, 2019

#1
+109563
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Let the long diagonal  of the rhombus    =  L

By the intersecting chord theorem  we have that

(3/2)r * (1/2)r    = [ ( 1/2) L ]^2

(3/4)r^2  = (1/4)L^2       multiply both sides by 4

(3)r^2  =  L^2       take the square root

(√3) r  = L

So.....the area of the rhombus  =   (1/2) product of the diagonals   =

(1/2) (r) (√3) r  =

[√3/2] r^2

Dec 27, 2019
edited by CPhill  Dec 27, 2019
#2
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Circle's radius      >  r = 2

Rhombus side      >  s = r

--II-- long diagonal > d1 =?             d1 = sqrt[s2- (r/2)2]*2      d1 = 3.464

--II-- short diagonal > d2 = r

Rhombus  area       > A = ?             A = 1/2(d1*r)  or  A = s * sqrt(3)  or A = (d1*d2)/2

A = 3.464 u2

Dec 27, 2019
edited by Dragan  Dec 27, 2019
edited by Dragan  Dec 27, 2019