1. **Problem statement:** Given a circle with diameter CD, AE = 6 cm, and CE = 4 cm, find the radius length of the circle M. 2. **Recall the properties:** The diameter CD passes through the center M, so the radius is half the diameter. 3. **Use the Pythagorean theorem:** Since AE and CE are segments from points on the circle to point E inside the circle, and AE and CE are perpendicular (horizontal and vertical), triangle AEC is right-angled at E. 4. Calculate AC using the Pythagorean theorem: $$AC = \sqrt{AE^2 + CE^2} = \sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13}$$ 5. Since AC is the radius of the circle (distance from center M to point A or C), the radius length is: $$r = 2\sqrt{13} \approx 7.21 \text{ cm}$$ 6. However, the options given are (a) 9, (b) 4.5, (c) 6, (d) 6.5. None exactly matches 7.21. 7. Re-examining the problem, since CD is diameter, and CE = 4 cm is part of radius, AE = 6 cm is another segment, the radius is the hypotenuse of triangle AEC, so radius = 7.21 cm. 8. The closest option is (a) 9 cm, but since none matches exactly, the radius length is approximately 7.21 cm. **Final answer:** Radius length of the circle M is approximately $7.21$ cm.