Using the standard formula for the area of a sector, (θ)/(360^°) × π r², we substitute θ = 36^°, π = (22)/(7), and r = (35)/(2) mm. This simplifies to (1)/(10) × (22)/(7) × (1225)/(4) = (385)/(4) mm², or 96.25 mm².

Using the standard formula for the area of a sector, (θ)/(360^°) × π r², we substitute θ = 36^°, π = (22)/(7), and r = (35)/(2) mm. This simplifies to (1)/(10) × (22)/(7) × (1225)/(4) = (385)/(4) mm², or 96.25 mm².

Q 9. A brooch is made with silver wire in the form of a circle with diameter $35\text{ mm}$ . The wire is also used in making $5$ diameters which divide the circle into $10$ equal sectors as shown in the figure. [Figure description: A circular brooch divided into $10$ equal sectors by $5$ straight line diameters passing through the center. Each sector contains decorative circular dots/elements.] Find (i) the total length of the silver wire required. (ii) the area of each sector of the brooch.

Answer: (i) Total length of silver wire required = $285\text{ mm}$ , (ii) Area of each sector = $\frac{385}{4}\text{ mm}^2$ (or $96.25\text{ mm}^2$ )

Step-by-step solution

Step 1: Find Circumference and Wire for 5 Diameters

The diameter of the circular brooch is $d = 35\text{ mm}$ , which gives a radius of $r = \frac{35}{2}\text{ mm}$ . The silver wire is used to form the outer circular boundary as well as $5$ diameters inside the circle. The circumference of the circular boundary is $\pi d = \frac{22}{7} \times 35 = 110\text{ mm}$ , and the total length needed for the $5$ diameters is $5 \times 35 = 175\text{ mm}$ .

Step 2: Calculate Total Length of Silver Wire

To obtain the total length of silver wire required, we add the circumference of the circle and the combined length of the $5$ diameters. This gives $110 + 175 = 285\text{ mm}$ .

Step 3: Determine Sector Angle

The $5$ diameters divide the entire circle into $10$ equal sectors. Since the complete angle at the centre of a circle is $360^\circ$ , the central angle subtended by each sector is $\theta = \frac{360^\circ}{10} = 36^\circ$ .

Step 4: Calculate Area of Each Sector

Using the standard formula for the area of a sector, $\frac{\theta}{360^\circ} \times \pi r^2$ , we substitute $\theta = 36^\circ$ , $\pi = \frac{22}{7}$ , and $r = \frac{35}{2}\text{ mm}$ . This simplifies to $\frac{1}{10} \times \frac{22}{7} \times \frac{1225}{4} = \frac{385}{4}\text{ mm}^2$ , or $96.25\text{ mm}^2$ .

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