1. Problem: Find the missing lengths in circle M given BD = 3, KM = 6, KP = 2\sqrt{7}, and some segment lengths. 2. Given AP = 2\sqrt{7} and CD = 3, we look to find AK, MD, AM, DS, KL, MP. 3. Typically, in circles, lengths involving chords, radii, and segments can be found using the Pythagorean theorem and properties such as perpendicular bisectors. 4. For B: Radius OB \perp AC at G means OG is a perpendicular segment from center to chord AC, so OG bisects AC. 5. 1) AG = 24 cm, so AC = 2 \times AG = 48 cm. 6. 2) AC = 38 cm, so G bisects AC, CG = \frac{38}{2} = 19 cm. 7. 3) OA=5 cm, OG=3 cm, triangle OAG right angled at G; by Pythagoras: $$AG = \sqrt{OA^2 - OG^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \text{ cm}$$ 8. 4) OG=5 cm, OC=13 cm, triangle OCG right angled at G: $$CG = \sqrt{OC^2 - OG^2} = \sqrt{13^2-5^2} = \sqrt{169-25} = \sqrt{144} = 12 \text{ cm}$$ 9. 5) CG=15 cm, OG=8 cm, triangle OCG: $$OC = \sqrt{CG^2 + OG^2} = \sqrt{15^2 + 8^2} = \sqrt{225+64} = \sqrt{289} = 17 \text{ cm}$$ Since FC is 13 cm (given), and OC and FC together form the chord or segment, further info needed for FC portion (or problem confirms FC=13). 10. 6) AC=16 cm, OG=15 cm, find GB. Because OG is perpendicular bisector, AG=GC=8 cm. Triangle OGB right angled, with OG=15 cm, GB unknown, OG perpendicularly bisects AC. If GB is segment on AC, since AG=8 and OG=15, by Pythagoras: $$GB = \sqrt{OG^2 - AG^2} = \sqrt{15^2 - 8^2} = \sqrt{225 - 64} = \sqrt{161} \approx 12.69 \text{ cm}$$ 11. 7) OG=6 cm, AC=16 cm so AG=GC=8 cm. Find BG is length from B to G. Triangle OBG right angled with OG=6 cm and OB radius. Without OB, cannot calculate BG directly unless OB known. 12. 8) BG=2 cm, OC=10 cm, find AG. If OG is perpendicular bisector, and OB is radius, then by Pythagoras: Calculate OG: $$OG = \sqrt{OC^2 - CG^2}$$ But CG unknown, similarly AG can be derived. Insufficient data without OG given in this part. 13. 9) OE=12 cm, OG=9 cm, find AC. Assuming similar triangles and chord bisectors, AC = 2 \times AG. AG calculated by: $$AG = \sqrt{OE^2 - OG^2} = \sqrt{12^2 - 9^2} = \sqrt{144 - 81} = \sqrt{63} = 3\sqrt{7} \text{ cm}$$ Thus, $$AC = 2 \times 3\sqrt{7} = 6\sqrt{7} \text{ cm}$$ 14. 10) FC = 34 cm, AC = 30 cm, find OG. Using Pythagoras in right triangles involving circle center and chords: If OG bisects AC, OG calculated as: $$OG = \sqrt{OC^2 - CG^2}$$ More data needed to calculate OG exactly here. Summary: Used circle chord properties and perpendicular bisector theorems. Final lengths found for problems with sufficient data are listed above. For the chord length or segments, use Pythagorean theorem as shown in example 7 and 9.