1. **Problem (a):** Two chords intersect inside a circle at point H. Given segments HK = 10.5, HJ = 28, and HN = 21, find HM. 2. **Formula:** When two chords intersect inside a circle, the products of the segments of each chord are equal: $$HK \times HJ = HM \times HN$$ 3. **Substitute known values:** $$10.5 \times 28 = HM \times 21$$ 4. **Calculate the left side:** $$294 = HM \times 21$$ 5. **Solve for HM:** $$HM = \frac{294}{21}$$ 6. **Simplify the fraction:** $$HM = \frac{\cancel{294}^{14} \times 21}{\cancel{21} \times 1} = 14$$ 7. **Answer (a):** $$HM = 14$$ 8. **Problem (b):** A tangent and a secant are drawn from an exterior point V to a circle. Given VG = 15 (tangent segment) and VD = 22.5 (secant segment), find CD. 9. **Formula:** The tangent-secant theorem states: $$VG^2 = VD \times VC$$ 10. **Note:** The secant segment VD is the entire length from V to D, and VC is the external part from V to C. Since CD is the segment inside the circle, we have: $$VD = VC + CD$$ 11. **Substitute known values:** $$15^2 = 22.5 \times VC$$ 12. **Calculate left side:** $$225 = 22.5 \times VC$$ 13. **Solve for VC:** $$VC = \frac{225}{22.5} = 10$$ 14. **Find CD:** $$CD = VD - VC = 22.5 - 10 = 12.5$$ 15. **Answer (b):** $$CD = 12.5$$