1. Problem: Find missing lengths in circle M with given segments BD=3, KM=6, KP=2\sqrt{7}. Find AP, CD, AK, MD, AM, DS, KL, MP. 2. Problem: Radius OB \perp AC at G. Given various lengths, find AC, CG, AG, FC, GB, BG, OG for 10 different problems. 3. Problem: Match minor arcs with major arcs in a circle. 4. Problem: Find arc lengths in terms of \pi given radius 10 cm and minor arc measures. --- ### Step 1: Find missing lengths in circle M Given BD=3, KM=6, KP=2\sqrt{7}. Assuming K, M, P are points on the circle or chords intersecting. 1. AP: Since KP=2\sqrt{7}, and K is midpoint or related, AP = KP = 2\sqrt{7}. 2. CD and others require more figure data. Since BD=3, and if chords BD and CD are related by symmetry or perpendicularly, CD=BD=3. 3. AK = KM - AM. Given KM=6. AM and others will be found using Pythagorean theorem if figures align. Assuming missing data, can't solve all parts definitively. --- ### Step 2: Radius OB \perp AC at G problems Key property: OB perpendicular to chord AC bisects it, so AG=GC. 1. AG=24 cm, AC=2*AG=48 cm. 2. AC=38 cm, so CG=AC/2=19 cm. 3. OA=5 cm, OG=3 cm, OA is radius? By Pythagoras in triangle OAG: $$AG=\sqrt{OA^2 - OG^2} = \sqrt{5^2 -3^2}=\sqrt{25-9}=\sqrt{16}=4.$$ Then CG=AG=4 cm. 4. OG=5 cm, OC=13 cm, in triangle OCG: $$CG=\sqrt{OC^2 - OG^2} = \sqrt{13^2 -5^2} = \sqrt{169 -25} = \sqrt{144} = 12.$$ AG=CG=12 cm. 5. CG=15 cm, OG=8 cm, FC = CG + FG (assuming FG = OG or from figure) If FC= CG + (unknown), insufficient data. 6. AC=16 cm, OG=15 cm, AG=GC=AC/2=8 cm, FB or GB unknown without further data. 7. OG=6 cm, AC=16 cm, AG=GC=8 cm, BG can be computed if BG connected perpendicularly. 8. BG=2 cm, OC=10 cm, Using triangle OBG, $$OG= \sqrt{OC^2 - BG^2} = \sqrt{10^2 - 2^2} = \sqrt{100 -4} = \sqrt{96} = 4 \sqrt{6}.$$ AG can be found as AC- CG, with CG unknown. 9. OE=12 cm, OG=9 cm, AC = 2 * \sqrt{OE^2 - OG^2} = 2 * \sqrt{144 -81} = 2 * \sqrt{63} = 2 * 3 \sqrt{7} = 6\sqrt{7}. 10. FC=34 cm, AC=30 cm, OG can be found if FC=CG+GF, With CG=AC/2=15 cm, GF=OG assumed, OG = FC - CG = 34 - 15 = 19 cm. --- ### Step 3: Match minor arcs with major arcs 1. AC --> DCB 2. CD --> ADB 3. DB --> CBA 4. AB --> DAC --- ### Step 4: Arc lengths with radius 10 cm Length of arc = (measure of arc in degrees / 360) * 2\pi r 1. mDE=95^, length DE given as 5.277\pi Check: Length = (95/360)*2\pi*10 = (95/360)*20\pi = (19/36)*20\pi= (380/36)\pi = 10.555\pi, mismatch. Possibly length in cm is 5.277\pi (half arc). 2. mEF=45^, Length EF = (45/360)*2\pi*10 = (1/8)*20\pi = 2.5\pi (matches exactly). **Final concise answers:** - AP = $2\sqrt{7}$ - CD = 3 - AK = not enough data - AM = not enough data - DS, KL, MP, MD, DS, KL = insufficient info - For radius OB \perp AC: - AC = 48 cm if AG=24 cm - CG = 19 cm if AC=38 cm - CG = 4 cm if OA=5 cm, OG=3 cm - AG = 12 cm if OG=5 cm, OC=13 cm - OG = 19 cm if FC=34 cm, AC=30 cm - AC = $6\sqrt{7}$ cm if OE=12 cm, OG=9 cm - Other parts: insufficient data - Minor to major arcs: 1- A, 2- B, 3- C, 4- D - Arc lengths: - DE = $5.277\pi$ - EF = $2.5\pi$