1. **Problem 1:** Given a circle with center $O$, radius $OJ = 10$, and an intercepted arc $JK$ of $120^\circ$, find the length of chord $JK$. 2. **Formula:** The length of a chord subtending an arc of angle $\theta$ in a circle of radius $r$ is given by: $$ JK = 2r \sin\left(\frac{\theta}{2}\right) $$ 3. **Apply the formula:** Here, $r = 10$ and $\theta = 120^\circ$. $$ JK = 2 \times 10 \times \sin\left(\frac{120^\circ}{2}\right) = 20 \sin(60^\circ) $$ 4. **Evaluate $\sin(60^\circ)$:** $$ \sin(60^\circ) = \frac{\sqrt{3}}{2} $$ 5. **Calculate $JK$:** $$ JK = 20 \times \frac{\sqrt{3}}{2} = 10\sqrt{3} $$ --- 6. **Problem 2:** Given a circle with center $O$, vertical diameter $HG$ passing through $E$, $OE = 8\sqrt{3}$, and an intercepted arc $HF$ of $150^\circ$, find the length of chord $HG$. 7. **Note:** Since $HG$ is a diameter, its length is twice the radius. We need to find the radius first. 8. **Find radius $r$:** Since $E$ lies on the radius $OE$, and $OE = 8\sqrt{3}$, the radius is $r = OE = 8\sqrt{3}$. 9. **Length of diameter $HG$:** $$ HG = 2r = 2 \times 8\sqrt{3} = 16\sqrt{3} $$ **Final answers:** - $JK = 10\sqrt{3}$ - $HG = 16\sqrt{3}$