# If OABC is a rhombus whose three vertices A,B and C lie on a circle with centre o. if the radius of the circle is 10cm.then the area of rhombus is

1
by mohen

2015-04-23T04:52:08+05:30

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I believe we need one more piece of information (data) to be able to define Rhobus uniquely.  We need the side and an angle (interior) or  radius and side or the two diagonals.

Here then we assume that the side  is  a  cm.      Radius of circumcircle of triangle ABC = R = 10 cm.

Let p = diagonal AC.    and  q = diagonal AO.      Z is the circumcenter.

R =  1/4 * a * a * p  / Area of ABC  = 1/4 * a² p / [ p q / 4 ]
= a² / q
=>    q = a² / R       --- (1)

In a rhombus we have
4 a² = p² + q²  = p² + (a²/R)²
=>    p = a/R * √[4 R² - a² ]    -- (2)

Area of Rhombus =  p q / 2 = 1/2 * a³ / R² * √[ 4 R² - a² ]
=  1/200 * a³ * √[400 - a²]      cm²

If we are given the angle inside Rhombus, or its side, or its diagonal then we can  uniquely define the Rhombus.

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