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प्रश्न
The area of the incircle of an equilateral triangle of side 42 cm is
पर्याय
\[22\sqrt{3} c m^2\]
231 cm2
462 cm2
924 cm2
उत्तर
Let ABC be the equilateral triangle such that AB = BC = CA = 42 cm. Also, let O be the centre and r be the radius of its incircle.
AB, BC and CA are tangents to the circle at M, N and P.
`∴ OM=ON=OP=r`
Area of ΔABC = Area (ΔOAB) + Area (ΔOBC) + Area (ΔOCA)
`⇒ sqrt3/4(42)^2=1/2xxrxxAB+1/2xxrxxBC+1/2xxrxx CA`
`⇒ sqrt3/4xx42xx42=1/2 r(AB+BC+CA)`
`⇒ 441sqrt3=1/2xxrxx(42+42+42)`
`⇒441sqrt3=1/2xxrxx(126)`
`⇒441sqrt3=63r`
`⇒r=(441sqrt3)/63`
`⇒ r=7sqrt3 cm`
Area of the circle `= pir^2=22/7 (7sqrt3)^2=22/7xx147=462 cm^2`
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