1. **Problem statement:** Find the locus of the midpoints of all chords of length 6 inches in a circle of radius 5 inches. 2. **Key concept:** The midpoint of a chord lies on a line perpendicular to the chord passing through the center of the circle. 3. **Formula:** For a chord of length $c$ in a circle of radius $r$, the distance $d$ from the center to the chord satisfies $$d = \sqrt{r^2 - \left(\frac{c}{2}\right)^2}$$ 4. **Calculate $d$:** Given $r=5$ and $c=6$, $$d = \sqrt{5^2 - \left(\frac{6}{2}\right)^2} = \sqrt{25 - 9} = \sqrt{16} = 4$$ 5. **Interpretation:** The midpoint of each chord lies exactly $d=4$ inches from the center. 6. **Locus:** The set of all such midpoints forms a circle centered at the center of the original circle with radius $4$ inches. **Final answer:** The locus of the midpoints of all chords of length 6 inches in a circle of radius 5 inches is a circle of radius 4 inches centered at the center of the original circle.