1. **Stating the problem:** We are given a circle with diameter BC perpendicular to chord DE at point F. We know BC = 12 inches, DE = 10 inches, AC = 6 inches, DA = 7.1 inches, FE = 5 inches, and FA = 5 inches. We need to analyze the relationships and possibly find missing lengths or verify properties. 2. **Important properties and formulas:** - Diameter BC is perpendicular to chord DE, so it bisects DE at F. - Since BC is diameter, the radius is half of BC: $$r = \frac{12}{2} = 6$$ inches. - The perpendicular from the center of a circle to a chord bisects the chord. 3. **Check if F is midpoint of DE:** Given FE = 5 inches and FA = 5 inches, since FE + ED = DE = 10 inches, and FE = 5, then FD = 5, so F is midpoint of DE. 4. **Check distances along diameter BC:** Given AC = 6 inches and DA = 7.1 inches, and since BC = 12 inches, point A lies on BC such that AC + DA = 6 + 7.1 = 13.1 inches, which is more than BC. This suggests A is not between B and C or there is a misinterpretation. 5. **Using Pythagoras theorem for triangle BFE:** Since BC is perpendicular to DE at F, triangle BFE is right-angled at F. - BF is radius minus FA: BF = 6 - 5 = 1 inch. - FE = 5 inches. - Using Pythagoras theorem: $$BE = \sqrt{BF^2 + FE^2} = \sqrt{1^2 + 5^2} = \sqrt{1 + 25} = \sqrt{26} \approx 5.1$$ inches. 6. **Summary:** - Radius of circle is 6 inches. - F is midpoint of chord DE. - Length BE is approximately 5.1 inches. Final answer: The radius of the circle is 6 inches, and the chord DE is bisected by the diameter BC at F, with each half measuring 5 inches.