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प्रश्न
ABC is an isosceles right-angled triangle with ∠ABC = 90°. A semi-circle is drawn with AC as the diameter. If AB = BC = 7 cm, find the area of the shaded region. [Take π = 22/7]
उत्तर
In ΔABC, using Pythagoras theorem
AC2 = AB2 + BC2
AC2 = (7)2 + (7)2
AC2 = 49 + 49
AC2 = 98
⇒ AC = `7sqrt2`
Radius of semi-circle = `"AC"/2 = (7sqrt2)/2`
∴ Area of the shaded region = Area of semi-circle – Area of ΔABC
Area of semi-circle = `1/2 pir^2 = 1/2 pi((7sqrt2)/2)^2`
= `(98pi)/8 = (49pi)/4`
= `49/4`
= `49/4 xx 22/7`
= `77/2 "cm"^2`
Area of ΔABC = `1/2` x BC x AB
= `1/2` x 7 x 7
= `49/2 "cm"^2 `
Thus, Area of the shaded region = `77/2 "cm"^2 - 49/2 "cm"^2 = 28/2 "cm"^2 = 14 "cm"^2`.
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