Let's solve question 2a step by step! 🎉 **Step 1:** Imagine a circle with center O and a chord AB inside it. **Step 2:** We know the length of chord AB is 24.2 cm. **Step 3:** The perimeter of the shape AAOB is 52.2 cm. This shape is made by the points A, A (again), O, and B. **Step 4:** The perimeter AAOB means: length of chord AB + length of arc AOB = 52.2 cm. **Step 5:** So, length of arc AOB = 52.2 - 24.2 = 28 cm. **Step 6:** The circle's radius OA = OB (since O is center), and AB is a chord. **Step 7:** Use the formula for chord length: $$AB = 2r \sin(\theta/2)$$ where $r$ is radius, $\theta$ is angle AOB in radians. **Step 8:** Use formula for arc length: $$\text{arc AOB} = r \times \theta$$ (where $\theta$ in radians) **Step 9:** From step 5, $$r \theta = 28$$ **Step 10:** From step 7, $$24.2 = 2r \sin(\theta/2)$$ **Step 11:** From step 9, $$\theta = \frac{28}{r}$$ **Step 12:** Substitute in step 10: $$24.2 = 2r \sin\left(\frac{28}{2r}\right) = 2r \sin\left(\frac{14}{r}\right)$$ **Step 13:** Let's try a radius $r$ that works. Try $r=15$ cm: $$2 \times 15 = 30$$ $$\sin(14/15) = \sin(0.9333) \approx 0.803$$ So, $$2r \sin(14/r) = 30 \times 0.803 = 24.09 \approx 24.2$$ Great! **Step 14:** So, $r \approx 15$ cm. **Step 15:** Find $\theta$: $$\theta = \frac{28}{15} = 1.8667 \text{ radians}$$ **Step 16:** Convert radians to degrees: $$\theta = 1.8667 \times \frac{180}{\pi} \approx 107^{\circ}$$ 🎯 **Answer:** $\boxed{107^{\circ}}$ (angle AOB to the nearest degree) --- **Visual:** Add ➕ =