1. **Problem Statement:** We are given a circle with center C and points A, B, D, E, F on the circle. Inside the circle, triangle ABC has an angle at A measuring 109°. We have two expressions for the lengths of chords AB and CB: $$AB = 3x + 2$$ $$CB = 5x - 7$$ We are asked to solve for $x$ and find the length of $BC$. 2. **Formula and Rules:** Since AB and BC are chords in the circle and the problem gives an angle at A, we assume the triangle is valid and the lengths must satisfy the given expressions. 3. **Set up the equation:** Since the problem states the angle at A is 109°, and the two chord lengths are given, the problem provides the equation: $$3x + 2 = 5x - 7$$ This equation comes from the problem statement, likely equating two expressions for the same segment or using a property of the triangle. 4. **Solve for $x$:** Subtract $3x$ from both sides: $$2 = 2x - 7$$ Add 7 to both sides: $$9 = 2x$$ Divide both sides by 2: $$x = \frac{9}{2} = 4.5$$ 5. **Find length of $BC$:** Substitute $x=4.5$ into $BC = 5x - 7$: $$BC = 5(4.5) - 7 = 22.5 - 7 = 15.5$$ **Final answer:** $$x = 4.5$$ $$BC = 15.5$$ Note: The user mentioned BC = 54.5, but based on the given expressions and solving the equation, the correct length is 15.5.