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Question
In a circular table cover of radius 32 cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in the given figure. Find the area of the design (Shaded region). [Use Π = 22/7]
Solution
Radius (r) of circle = 32 cm
AD is the median of ΔABC.
`AO =2/3 AD = 32`
AD = 48 cm
In ΔABD,
AB2 = AD2 + BD2
`AB^2 = (48)^2 + ((AB)/2)^2`
`(3AB^2)/4 = (48)^2`
`AB = (48xx2)/sqrt3 = 96/sqrt3`
`= 32sqrt3 cm`
Area of equilateral triangle ΔABC = `sqrt3/4(32sqrt3)^2`
`=sqrt3/4 xx 32xx32xx2 = 96xx8xxsqrt3`
`= 768sqrt3 cm^2`
Area of circle = πr2
`= 22/7xx(32)^2`
`=22/7 xx 1024`
`= 22528/7 cm^2`
Area of design = Area of circle − Area of ΔABC
`= ((22528)/7 - 768sqrt3) cm^2`
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