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Areas Related to Circles
NCERT Exemplar Ch 10
CBSE_NCERT_EXEMPLAR_CH10
Grade 10
Question:
Find the area of the shaded region in figure, where a circular arc of radius $6\text{ cm}$ has been drawn with vertex $O$ of an equilateral triangle $OAB$ of side $12\text{ cm}$ as centre.
Step-by-Step Solution
Key Concept: Shaded area $= \text{Area of major sector of radius } 6\text{ cm} + \text{Area of equilateral } \Delta OAB$. Major sector angle $= 360^\circ - 60^\circ = 300^\circ$.
Stepwise Solution:
Angle of equilateral triangle $= 60^\circ$. Major sector angle $= 360^\circ - 60^\circ = 300^\circ$. [1.0 Mark]
Major sector area $= \dfrac{300}{360} \times \pi (6)^2 = \dfrac{5}{6} \times \dfrac{22}{7} \times 36 = \dfrac{660}{7}\text{ cm}^2$. [2.0 Marks]
Area of equilateral $\Delta OAB = \dfrac{\sqrt{3}}{4} (12)^2 = 36\sqrt{3}\text{ cm}^2$. [1.0 Mark]
Total shaded area $= \left(\dfrac{660}{7} + 36\sqrt{3}\right)\text{ cm}^2$. [1.0 Mark]
Marking Scheme:
• Major sector angle $300^\circ$: 1.0 Mark • Major sector area $= 660/7\text{ cm}^2$: 2.0 Marks • Equilateral triangle area $= 36\sqrt{3}\text{ cm}^2$: 1.0 Mark • Total shaded area $= 660/7 + 36\sqrt{3}\text{ cm}^2$: 1.0 Mark
Correct Answer:
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