1. **State the problem:** We have a circle with radius 20, and a right triangle inside it with legs $x$ and $y$. The equation $x^2 + y^2 = 20^2$ represents the Pythagorean theorem for the triangle formed by the radius and the legs. 2. **Given:** - $x^2 + y^2 = 20^2$ - $y^2 = 14^2 + 12^2$ 3. **Calculate $y^2$ using the chord segments:** $$y^2 = 14^2 + 12^2 = 196 + 144 = 340$$ 4. **Substitute $y^2$ into the first equation:** $$x^2 + 340 = 400$$ 5. **Solve for $x^2$:** $$x^2 = 400 - 340 = 60$$ 6. **Find $x$ by taking the square root:** $$x = \sqrt{60} = \sqrt{4 \times 15} = 2\sqrt{15}$$ 7. **Summary:** - $y = \sqrt{340} = \sqrt{14^2 + 12^2}$ - $x = 2\sqrt{15}$ **Final answer:** $$x = 2\sqrt{15}$$