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The boundary of the shaded region in the given diagram consists of five semicircular areas. If AB=7cm, BC=3.5cm, CD=7cm and DE=7 cm, then area of the shaded region is

A) \( \Large \frac{17.5}{2} \times \frac{3.5}{2}\ \pi\ cm^{2} \)
B) \( \Large \frac{17.5 \times 21}{2}\ \pi\ cm^{2} \)
C) \( \Large \frac{35 \times 21}{16}\ \pi\ cm^{2} \)
D) \( \Large \frac{35 \times 21 \pi }{8}\ cm^{2} \)

Correct Answer:



C) \( \Large \frac{35 \times 21}{16}\ \pi\ cm^{2} \)

Description for Correct answer:

Required area = (Area of semi circle AC - Area of semi-circle AB) + (Area of semi-circle BE - Area of semi circle CD) + (Area of semi-circle DE)

= \( \Large \left[ \frac{1}{2} \pi \left(\frac{10.5}{2}\right)^{2}-\frac{1}{2} \pi \left(\frac{7}{2}\right)^{2} \right] \) \( \Large + \left[ \frac{1}{2} \pi \left(\frac{17.5}{2}\right)^{2} - \frac{1}{2} \pi \left(\frac{7}{2}\right)^{2} \right] + \frac{1}{2} \pi \left(\frac{7}{2}\right)^{2} \)

= \( \Large \frac{ \pi }{2}\left[ \left(\frac{10.5}{2}\right)^{2}+ \left(\frac{17.5}{2}\right)^{2}- \left(\frac{7}{2}\right)^{2} \right] \)

= \( \Large \frac{ \pi }{8}\left[ \left(10.5\right)^{2}+ \left(17.5\right)^{2}- \left(7\right)^{2} \right] \)

= \( \Large \frac{ \pi }{8}\left[ \left(10.5\right)^{2}+ \left(10.5\right) \left(24.5\right) \right] \)

= \( \Large \frac{ \pi }{8}\left[ \left(10.5\right) \left(35\right) \right] \)

= \( \Large \frac{ \pi }{16} \left(35\right) \left(21\right) cm^{2} \)

Part of solved Loci and concurrency questions and answers : >> Elementary Mathematics >> Loci and concurrency

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