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.
Generated by claude-sonnet-4-6 · 2026-06-15 10:36 · grounding stimulus
Model Answer
Given: Diameter = 35 mm, so radius $r = \frac{35}{2}$ mm. The circle is divided into 10 equal sectors.
(i) Central angle of each sector:
$$\theta = \frac{360°}{10} = \mathbf{36°}$$
(ii) Length of arc ACB:
Arc ACB spans 2 sectors (as it covers 2 equal parts).
$$\text{Length} = \frac{2 \times 36}{360} \times 2\pi r = \frac{72}{360} \times 2 \times \frac{22}{7} \times \frac{35}{2}$$
$$= \frac{1}{5} \times 110 = \mathbf{22 \text{ mm}}$$
(iii) Area of each sector:
$$\text{Area} = \frac{\theta}{360°} \times \pi r^2 = \frac{36}{360} \times \frac{22}{7} \times \frac{35}{2} \times \frac{35}{2}$$
$$= \frac{1}{10} \times \frac{22}{7} \times \frac{1225}{4} = \frac{1}{10} \times \frac{26950}{28} = \frac{2695}{28} \approx \mathbf{96.25 \text{ mm}^2}$$
Source: Areas Related to Circles, Sector of a Circle
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Explanation
- Part (i): Simply divide 360° by the number of sectors (10).
- Part (ii): Arc ACB likely spans 2 sectors (72°) based on the figure label showing it as one prominent arc. Use the arc length formula $\frac{\theta}{360} \times 2\pi r$.
- Part (iii): Use the sector area formula $\frac{\theta}{360} \times \pi r^2$ for one sector (36°). Examiners expect clean substitution and simplification using $\pi = \frac{22}{7}$.