1. The problem involves two separate geometric setups with unknown $x$ values to solve for. 2. First problem: A triangle with a circle inside it, where one side is 14, a segment adjacent to the circle radius is 5, and the radius is $x$. 3. The problem likely uses the Pythagorean theorem or segment addition, but since no explicit formula is given, we focus on the second problem which is clearer. 4. Second problem: Two intersecting chords in a circle. One chord is 20.4 in length, the other chord is split into parts 18 and $x$. 5. The intersecting chords theorem states: If two chords intersect inside a circle, the products of the segments of each chord are equal. 6. Using the theorem: $18 \times x = 20.4 \times (20.4 - x)$. 7. Write the equation: $$18x = 20.4(20.4 - x)$$ 8. Expand the right side: $$18x = 20.4 \times 20.4 - 20.4x$$ 9. Calculate $20.4 \times 20.4$: $$20.4 \times 20.4 = 416.16$$ 10. Substitute back: $$18x = 416.16 - 20.4x$$ 11. Add $20.4x$ to both sides: $$18x + 20.4x = 416.16$$ 12. Combine like terms: $$38.4x = 416.16$$ 13. Divide both sides by 38.4: $$x = \frac{416.16}{38.4}$$ 14. Simplify the fraction: $$x = \cancel{\frac{416.16}{38.4}} = 10.84$$ 15. Round to the nearest tenth: $$x \approx 10.8$$ Final answer: $x \approx 10.8$