1. **Problem statement:** Given a circle with center O and radius 8 m, points P, Q, and R lie on the circumference. OPQ is a straight line, $\angle QOR = 36.9^\circ$, and OR is perpendicular to QR. We need to find: (i) the arc length PR, (ii) the perimeter of the shaded region PQR. 2. **Relevant formulas and rules:** - Arc length $s = r\theta$ where $r$ is radius and $\theta$ is the central angle in radians. - Perimeter of shaded region PQR = length of arc PR + length of PQ + length of QR. - Since OPQ is a straight line, $\angle POQ = 180^\circ$. - OR is perpendicular to QR means $\angle ORQ = 90^\circ$. 3. **Find the central angle $\angle POR$:** Since $\angle QOR = 36.9^\circ$ and OPQ is a straight line, $\angle POQ = 180^\circ$. Therefore, $\angle POR = \angle POQ - \angle QOR = 180^\circ - 36.9^\circ = 143.1^\circ$. 4. **Calculate arc length PR:** Convert $\angle POR$ to radians: $$\theta = 143.1^\circ \times \frac{\pi}{180} = \frac{143.1\pi}{180}$$ Arc length: $$s = r\theta = 8 \times \frac{143.1\pi}{180} = \frac{1144.8\pi}{180} = \frac{1144.8\pi}{180}$$ Simplify fraction: $$\frac{1144.8}{180} = 6.36$$ So, $$s = 6.36\pi \approx 19.97\text{ m}$$ 5. **Find lengths PQ and QR:** - Since OPQ is a straight line and O is center, P and Q lie on the circle with radius 8 m. - Length PQ = OP + OQ = 8 + 8 = 16 m. - To find QR, use right triangle ORQ where OR = 8 m (radius), and $\angle ORQ = 90^\circ$. - $\angle QOR = 36.9^\circ$ is the angle at O between Q and R. Using triangle ORQ: - QR is opposite to $\angle QOR = 36.9^\circ$, OR is adjacent. Calculate QR: $$QR = OR \times \tan(36.9^\circ) = 8 \times \tan(36.9^\circ)$$ Calculate $\tan(36.9^\circ)$: approximately 0.75 So, $$QR = 8 \times 0.75 = 6\text{ m}$$ 6. **Calculate perimeter of shaded region PQR:** $$\text{Perimeter} = \text{arc length PR} + PQ + QR = 19.97 + 16 + 6 = 41.97\text{ m}$$ **Final answers:** (i) Arc length PR $\approx 19.97$ m (ii) Perimeter of shaded region PQR $\approx 41.97$ m