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Case Study - 2: A brooch is a decorative piece often worn on clothing like jackets, blouses or dresses to add elegance. Made from precious metals and decorated with gemstones, brooches come in many shapes and designs. One such brooch is made with silver wire in the form of a circle with diameter $35$ mm. The wire is also used in making $5$ diameters which divide the circle into $10$ equal sectors as shown in the figure. (i) Find the central angle of each sector. (ii) Find the length of the arc $ACB$. (iii) (a) Find the area of each sector of the brooch. OR (iii) (b) Find the total length of the silver wire used.
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(i) Central angle $= \frac{360^\circ}{10} = 36^\circ$. (ii) Length of arc $ACB = \frac{1}{10} \times 2 \times \frac{22}{7} \times \frac{35}{2} = 11$ mm. (iii)(a) Area $= \frac{1}{10} \times \frac{22}{7} \times \frac{35}{2} \times \frac{35}{2} = 96.25$ mm$^2$. (iii)(b) Length of wire $= 2 \times \frac{22}{7} \times \frac{35}{2} + 5 \times 35 = 285$ mm.