1. **State the problem:** Given a circle with radius $OR = 18$ cm and central angle $\angle SOR = 155^\circ$, find: (a) the length of the minor arc $ST$ in cm, (b) the area of the shaded region in cm². 2. **Formulas and rules:** - Length of an arc: $$\text{Arc length} = r \times \theta$$ where $r$ is the radius and $\theta$ is the angle in radians. - Area of a sector: $$\text{Area} = \frac{1}{2} r^2 \theta$$ where $\theta$ is in radians. - To convert degrees to radians: $$\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}$$ - Given $\pi = \frac{22}{7}$. 3. **Calculate minor arc length $ST$:** - The minor arc corresponds to the smaller angle between $S$ and $T$. Since $\angle SOR = 155^\circ$, the minor arc angle is $360^\circ - 155^\circ = 205^\circ$ or the problem might consider the minor arc as $155^\circ$ itself (usually minor arc is the smaller angle, so $155^\circ$ is minor here). - We use $\theta = 155^\circ$. - Convert $155^\circ$ to radians: $$ \theta = 155 \times \frac{22}{7} \times \frac{1}{180} = \frac{155 \times 22}{7 \times 180} $$ - Simplify: $$ \theta = \frac{155 \times 22}{1260} = \frac{3410}{1260} = \frac{3410 \div 10}{1260 \div 10} = \frac{341}{126} $$ - Arc length: $$ L = r \times \theta = 18 \times \frac{341}{126} = \frac{18 \times 341}{126} $$ - Simplify fraction: $$ \frac{18}{126} = \frac{\cancel{18}^2}{\cancel{126}^7} \Rightarrow L = 2 \times 341 = 682 \text{ cm} $$ - This is too large, so re-check: actually $\frac{18}{126} = \frac{3}{21} = \frac{1}{7}$, so: $$ L = \frac{1}{7} \times 341 = 48.71 \text{ cm (approx)} $$ 4. **Calculate area of the sector (shaded region):** - Area formula: $$ A = \frac{1}{2} r^2 \theta = \frac{1}{2} \times 18^2 \times \frac{341}{126} = \frac{1}{2} \times 324 \times \frac{341}{126} $$ - Simplify: $$ A = 162 \times \frac{341}{126} = \frac{162 \times 341}{126} $$ - Simplify fraction $\frac{162}{126} = \frac{81}{63} = \frac{27}{21} = \frac{9}{7}$: $$ A = \frac{9}{7} \times 341 = 438.43 \text{ cm}^2 \text{ (approx)} $$ **Final answers:** (a) Length of minor arc $ST$ is approximately $48.71$ cm. (b) Area of the shaded region is approximately $438.43$ cm².