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Question
A circle is inscribed in an equilateral triangle ABC is side 12 cm, touching its sides (the following figure). Find the radius of the inscribed circle and the area of the shaded part.
Solution
We have to find the area of the shaded portion. We have`ΔABC`which is an equilateral triangle and`AB=12 cm`.
We have O as the incentre and OP, OQ and OR are equal.
So,
`ar(Δ ABC)=ar(ΔOAB)+ar(ΔOBC)+ae(ΔOCA)`
Thus,
`sqrt3/4(12)^2=3(1/2(12))(r)`
`r=(36sqrt3)/18 cm`
`=2sqrt3 cm`
So area of the shaded region,
`= ar (Δ ABC)-Area of the circle`
`=sqrt3/4(12)^2-22/7(2sqrt3)^2`
`=(62.35-37.71)cm^2`
`=24.64 cm^2`
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