GIÁO TRÌNH TOÁN 9

["GI\u00c1O TR\u00ccNH TO\u00c1N 9 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M B\u00e0i 36. Cho t\u1ee9 gi\u00e1c ABCD c\u00f3 A + C = 180\u25e6. E l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AD v\u00e0 BC k\u00e9o d\u00e0i. Ch\u1ee9ng minh r\u1eb1ng EA.EB = ED.EC. B\u00e0i 37. Cho t\u1ee9 gi\u00e1c ABCD c\u00f3 A + C = 180\u25e6. O l\u00e0 giao \u0111i\u1ec3m c\u1ee7a 2 \u0111\u01b0\u1eddng ch\u00e9o. Ch\u1ee9ng minh r\u1eb1ng OA.OC = OB.OD. B\u00e0i 38. Cho t\u1ee9 gi\u00e1c ABCD c\u00f3 CAD = CBD. G\u1ecdi O l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AC v\u00e0 DB. Ch\u1ee9ng minh r\u1eb1ng hai tam gi\u00e1c AOB v\u00e0 DOC \u0111\u1ed3ng d\u1ea1ng. B\u00e0i 39. Cho tam gi\u00e1c ABC l\u1ea5y D thu\u1ed9c AB v\u00e0 E thu\u1ed9c AC sao cho AE.AC = AB.AD. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c BDEC n\u1ed9i ti\u1ebfp. B\u00e0i 40. Cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH v\u00e0 trung tuy\u1ebfn BI. K l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a A l\u00ean BI. Ch\u1ee9ng minh r\u1eb1ng: 1) BK.BI = BH.BC; 2) T\u1ee9 gi\u00e1c HKIC n\u1ed9i ti\u1ebfp. B\u00e0i 41. Cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH. V\u1ebd \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m H b\u00e1n k\u00ednh HA, g\u1ecdi DE l\u00e0 \u0111\u01b0\u1eddng k\u00ednh b\u1ea5t k\u00ec c\u1ee7a (H). Ch\u1ee9ng minh r\u1eb1ng: 1) HB.HC = HD.HE; 2) 4 \u0111i\u1ec3m B, D, C, E c\u00f9ng thu\u1ed9c m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n. B\u00c0I T\u1eacP N\u00c2NG CAO B\u00e0i 42. Cho tam gi\u00e1c nh\u1ecdn ABC c\u00f3 H l\u00e0 tr\u1ef1c t\u00e2m v\u00e0 M l\u00e0 trung \u0111i\u1ec3m BC. K\u1ebb \u0111\u01b0\u1eddng th\u1eb3ng qua H vu\u00f4ng g\u00f3c v\u1edbi HM v\u00e0 c\u1eaft AB, AC l\u1ea7n l\u01b0\u1ee3t t\u1ea1i P v\u00e0 Q. Ch\u1ee9ng minh r\u1eb1ng M P = M Q. B\u00e0i 43. Cho h\u00ecnh vu\u00f4ng ABCD t\u00e2m O. Tr\u00ean c\u1ea1nh BC l\u1ea5y \u0111i\u1ec3m E, g\u1ecdi G l\u00e0 giao \u0111i\u1ec3m c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng AE v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng DC, g\u1ecdi F l\u00e0 giao \u0111i\u1ec3m c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng OE v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng BG. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c OBF C n\u1ed9i ti\u1ebfp. B\u00e0i 44. Cho tam gi\u00e1c ABC g\u1ecdi D l\u00e0 giao \u0111i\u1ec3m c\u1ee7a tia ph\u00e2n gi\u00e1c A v\u00e0 \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a BC. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c ABDC n\u1ed9i ti\u1ebfp. B\u00e0i 45. Cho h\u00ecnh vu\u00f4ng ABCD. L\u1ea5y M, N tr\u00ean c\u1ea1nh BC v\u00e0 CD sao cho M AN = 45\u25e6. G\u1ecdi E v\u00e0 F l\u1ea7n l\u01b0\u1ee3t l\u00e0 giao \u0111i\u1ec3m c\u1ee7a BD v\u1edbi AM v\u00e0 AN, I l\u00e0 giao \u0111i\u1ec3m EN v\u00e0 F M. Ch\u1ee9ng minh r\u1eb1ng M N = AI. B\u00e0i 46. Cho h\u00ecnh vu\u00f4ng ABCD t\u00e2m O. Tr\u00ean \u0111o\u1ea1n OB l\u1ea5y I, \u0111\u01b0\u1eddng th\u1eb3ng AI c\u1eaft BC t\u1ea1i E, qua I k\u1ebb \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi AE c\u1eaft AB t\u1ea1i M v\u00e0 c\u1eaft DC t\u1ea1i N. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c AM EN l\u00e0 h\u00ecnh thang c\u00e2n. B\u00e0i 47. Tr\u00ean c\u00f9ng n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n \u0111\u01b0\u1eddng k\u00ednh AB l\u1ea5y \u0111i\u1ec3m C v\u00e0 E. V\u1ebd \u0111\u01b0\u1eddng tr\u00f2n (A; AC) v\u00e0 (B; BC) c\u1eaft nhau t\u1ea1i \u0111i\u1ec3m th\u1ee9 hai l\u00e0 D. V\u1ebd tia \u0111\u1ed1i c\u1ee7a tia AE c\u1eaft (A) t\u1ea1i G, tia \u0111\u1ed1i c\u1ee7a tia BE c\u1eaft (B) t\u1ea1i F. Ch\u1ee9ng minh r\u1eb1ng ba \u0111i\u1ec3m G, D, F th\u1eb3ng h\u00e0ng. \u0110\u01af\u1edcNG TR\u00d2N EULER | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 95","| C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT | Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m B\u00e0i 48. Cho tam gi\u00e1c ABC c\u00f3 ba \u0111\u01b0\u1eddng cao AD, BE, CF c\u1eaft nhau t\u1ea1i H. G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BH. Ch\u1ee9ng minh r\u1eb1ng T\u1ee9 gi\u00e1c F EDM n\u1ed9i ti\u1ebfp. B\u00e0i 49. Cho tam gi\u00e1c ABC c\u00f3 ba \u0111\u01b0\u1eddng cao AD, BE, CF c\u1eaft nhau t\u1ea1i H. G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BC. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c EF DM n\u1ed9i ti\u1ebfp. B\u00e0i 50. Cho tam gi\u00e1c ABC c\u00f3 ba \u0111\u01b0\u1eddng cao AD, BE, CF c\u1eaft nhau t\u1ea1i H. G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BC, J l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BH. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c EF M J n\u1ed9i ti\u1ebfp. B\u00e0i 51. Cho tam gi\u00e1c ABC c\u00f3 ba \u0111\u01b0\u1eddng cao AD, BE, CF c\u1eaft nhau t\u1ea1i H. G\u1ecdi M, I, J l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BH, CH, AH. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c EM IJ n\u1ed9i ti\u1ebfp. B\u00e0i 52. Cho tam gi\u00e1c ABC c\u00f3 ba \u0111\u01b0\u1eddng cao AD, BE, CF c\u1eaft nhau t\u1ea1i H. G\u1ecdi M, I, J, K l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BH, CH, AH v\u00e0 AB. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c KJIM n\u1ed9i ti\u1ebfp. B\u00e0i 53. Cho h\u00ecnh vu\u00f4ng ABCD t\u00e2m O. Tr\u00ean AB v\u00e0 BC l\u1ea7n l\u01b0\u1ee3t l\u1ea5y c\u00e1c \u0111i\u1ec3m F v\u00e0 E sao cho AF = BE, g\u1ecdi M l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AE v\u00e0 DF. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c M OEB n\u1ed9i ti\u1ebfp. B\u00e0i 54. Cho \u25b3ABC vu\u00f4ng \u1edf A. Tr\u00ean AC l\u1ea5y \u0111i\u1ec3m M v\u00e0 v\u1ebd \u0111\u01b0\u1eddng tr\u00f2n \u0111\u01b0\u1eddng k\u00ednh M C. K\u1ebb BM c\u1eaft \u0111\u01b0\u1eddng tr\u00f2n t\u1ea1i D. \u0110\u01b0\u1eddng th\u1eb3ng DA c\u1eaft \u0111\u01b0\u1eddng tr\u00f2n t\u1ea1i S. Ch\u1ee9ng minh r\u1eb1ng: 1) T\u1ee9 gi\u00e1c ABCD n\u1ed9i ti\u1ebfp; 2) ABD = ACD; 3) CA l\u00e0 ph\u00e2n gi\u00e1c c\u1ee7a SCB. B\u00e0i 55. Cho t\u1ee9 gi\u00e1c ABCD n\u1ed9i ti\u1ebfp n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n \u0111\u01b0\u1eddng k\u00ednh AD. Hai \u0111\u01b0\u1eddng ch\u00e9o AC v\u00e0 BD c\u1eaft nhau t\u1ea1i E. V\u1ebd EF vu\u00f4ng g\u00f3c v\u1edbi AD. 1) Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c ABEF, t\u1ee9 gi\u00e1c DCEF n\u1ed9i ti\u1ebfp; 2) Ch\u1ee9ng minh r\u1eb1ng CA l\u00e0 ph\u00e2n gi\u00e1c c\u1ee7a BCF ; 3) G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a DE. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c BcmF n\u1ed9i ti\u1ebfp. B\u00e0i 56. T\u1ee9 gi\u00e1c ABCD n\u1ed9i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n \u0111\u01b0\u1eddng k\u00ednh AD. Hai \u0111\u01b0\u1eddng ch\u00e9o AC, BD c\u1eaft nhau t\u1ea1i E. H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c c\u1ee7a E tr\u00ean AD l\u00e0 F. \u0110\u01b0\u1eddng th\u1eb3ng CF c\u1eaft \u0111\u01b0\u1eddng tr\u00f2n t\u1ea1i \u0111i\u1ec3m th\u1ee9 hai l\u00e0 M. Giao \u0111i\u1ec3m c\u1ee7a BD v\u00e0 CF l\u00e0 N. Ch\u1ee9ng minh r\u1eb1ng: 1) CEF D l\u00e0 t\u1ee9 gi\u00e1c n\u1ed9i ti\u1ebfp; 2) F A l\u00e0 ph\u00e2n gi\u00e1c c\u1ee7a BF M . 3) BE.DN = EN.BD. B\u00e0i 57. Cho t\u1ee9 gi\u00e1c ABCD n\u1ed9i ti\u1ebfp n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n \u0111\u01b0\u1eddng k\u00ednh AD. Hai \u0111\u01b0\u1eddng ch\u00e9o AC v\u00e0 BD c\u1eaft nhau t\u1ea1i E. V\u1ebd EF vu\u00f4ng g\u00f3c v\u1edbi AD. G\u1ecdi M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a DE. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c BcmF n\u1ed9i ti\u1ebfp. B\u00e0i 58. Cho \u25b3ABC (A = 90\u25e6; AB > AC) v\u00e0 m\u1ed9t \u0111i\u1ec3m M n\u1eb1m tr\u00ean \u0111o\u1ea1n AC (M kh\u00f4ng tr\u00f9ng v\u1edbi A v\u00e0 C). G\u1ecdi N v\u00e0 D l\u1ea7n l\u01b0\u1ee3t l\u00e0 giao \u0111i\u1ec3m th\u1ee9 hai c\u1ee7a BC v\u00e0 M B v\u1edbi \u0111\u01b0\u01a1ng tr\u00f2n \u0111\u01b0\u1eddng 96 N\u0103m h\u1ecdc: 2023 - 2024","S GI\u00c1O TR\u00ccNH TO\u00c1N 9 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M k\u00ednh M C; g\u1ecdi S l\u00e0 giao \u0111i\u1ec3m th\u1ee9 hai gi\u1eefa AD v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n \u0111\u01b0\u1eddng k\u00ednh M C; T l\u00e0 giao \u0111i\u1ec3m c\u1ee7a M N v\u00e0 AB. Ch\u1ee9ng minh r\u1eb1ng: 1) B\u1ed1n \u0111i\u1ec3m A, M, N v\u00e0 B c\u00f9ng thu\u1ed9c m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n; 2) cm l\u00e0 ph\u00e2n gi\u00e1c c\u1ee7a BCS; TA TC 3) = . TD TB B\u00e0i 59. Cho \u0111\u01b0\u1eddng tr\u00f2n (O) v\u00e0 \u0111i\u1ec3m A n\u1eb1m ngo\u00e0i \u0111\u01b0\u1eddng tr\u00f2n. Qua A d\u1ef1ng hai ti\u1ebfp tuy\u1ebfn AM v\u00e0 AN v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n (M, N l\u00e0 c\u00e1c ti\u1ebfp \u0111i\u1ec3m) v\u00e0 m\u1ed9t c\u00e1t tuy\u1ebfn b\u1ea5t k\u00ec c\u1eaft \u0111\u01b0\u1eddng tr\u00f2n t\u1ea1i P, Q. G\u1ecdi L l\u00e0 trung \u0111i\u1ec3m c\u1ee7a P Q. 1) Ch\u1ee9ng minh r\u1eb1ng 5 \u0111i\u1ec3m O; L; M ; A; N c\u00f9ng thu\u1ed9c m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n; 2) Ch\u1ee9ng minh r\u1eb1ng LA l\u00e0 ph\u00e2n gi\u00e1c c\u1ee7a M LN ; 3) G\u1ecdi I l\u00e0 giao \u0111i\u1ec3m c\u1ee7a M N v\u00e0 LA. Ch\u1ee9ng minh r\u1eb1ng M A2 = AI.AL; 4) G\u1ecdi K l\u00e0 giao \u0111i\u1ec3m c\u1ee7a M L v\u1edbi (O). Ch\u1ee9ng minh r\u1eb1ng KN\/\/AQ; 5) Ch\u1ee9ng minh r\u1eb1ng \u25b3KLN c\u00e2n. B\u00e0i 60. Cho \u0111\u01b0\u1eddng tr\u00f2n (O; R) ti\u1ebfp x\u00fac v\u1edbi \u0111\u01b0\u1eddng th\u1eb3ng d t\u1ea1i A. Tr\u00ean d l\u1ea5y \u0111i\u1ec3m H kh\u00f4ng tr\u00f9ng v\u1edbi \u0111i\u1ec3m A v\u00e0 AH < R. Qua H k\u1ebb \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi d, \u0111\u01b0\u1eddng th\u1eb3ng n\u00e0y c\u1eaft \u0111\u01b0\u1eddng tr\u00f2n t\u1ea1i hai \u0111i\u1ec3m E v\u00e0 B (E n\u1eb1m gi\u1eefa B v\u00e0 H). 1) Ch\u1ee9ng minh r\u1eb1ng ABE = EAH v\u00e0 \u25b3ABH \u25b3EAH; 2) L\u1ea5y \u0111i\u1ec3m C tr\u00ean d sao cho H l\u00e0 trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n AC, \u0111\u01b0\u1eddng th\u1eb3ng CE c\u1eaft AB t\u1ea1i K. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c AHEK l\u00e0 t\u1ee9 gi\u00e1c n\u1ed9i ti\u1ebfp; \u221a 3) X\u00e1c \u0111\u1ecbnh v\u1ecb tr\u00ed \u0111i\u1ec3m H \u0111\u1ec3 AB = R 3. B\u00e0i 61. Cho \u25b3ABC kh\u00f4ng c\u00e2n, \u0111\u01b0\u1eddng cao AH, n\u1ed9i ti\u1ebfp trong \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O. G\u1ecdi E, F th\u1ee9 t\u1ef1 l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a B, C l\u00ean \u0111\u01b0\u1eddng k\u00ednh AD c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n (O) v\u00e0 M, N, K th\u1ee9 t\u1ef1 l\u00e0 trung \u0111i\u1ec3m c\u1ee7a BC, AB, EC. Ch\u1ee9ng minh r\u1eb1ng: 1) B\u1ed1n \u0111i\u1ec3m A, B, H, E c\u00f9ng n\u1eb1m tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m N v\u00e0 HE\/\/CD; 2) M N l\u00e0 \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a HE, M K l\u00e0 \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a ED; 3) \u0110i\u1ec3m M c\u00e1ch \u0111\u1ec1u H, E, F. B\u00e0i 62. Cho \u25b3ABC kh\u00f4ng c\u00e2n, \u0111\u01b0\u1eddng cao AH, n\u1ed9i ti\u1ebfp trong \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O. G\u1ecdi E, F th\u1ee9 t\u1ef1 l\u00e0 h\u00ecnh chi\u1ebfu c\u1ee7a B, C l\u00ean \u0111\u01b0\u1eddng k\u00ednh AD c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n (O) v\u00e0 M, N th\u1ee9 t\u1ef1 l\u00e0 trung \u0111i\u1ec3m | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 97","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m c\u1ee7a BC, AB. Ch\u1ee9ng minh r\u1eb1ng: | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT 1) B\u1ed1n \u0111i\u1ec3m A, B, H, E c\u00f9ng n\u1eb1m tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m N. 2) HE\/\/CD; 3) M l\u00e0 t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp \u25b3HEF . B\u00e0i 63. Cho hai ti\u1ebfp tuy\u1ebfn t\u1ea1i B v\u00e0 C c\u1ee7a (O) c\u1eaft nhau t\u1ea1i A, OA c\u1eaft BC t\u1ea1i H. V\u1ebd c\u00e1t tuy\u1ebfn ADE v\u1edbi (O) sao cho DB < DC. Ch\u1ee9ng minh r\u1eb1ng: 1) T\u1ee9 gi\u00e1c OHDE n\u1ed9i ti\u1ebfp; 2) HB l\u00e0 tia ph\u00e2n gi\u00e1c c\u1ee7a EHD. B\u00e0i 64. Cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A c\u00f3 \u0111\u01b0\u1eddng cao AH. G\u1ecdi I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AH, K l\u00e0 trung \u0111i\u1ec3m c\u1ee7a CH, \u0111\u01b0\u1eddng th\u1eb3ng BI c\u1eaft AK t\u1ea1i E. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c IEKH l\u00e0 t\u1ee9 gi\u00e1c n\u1ed9i ti\u1ebfp. B\u00e0i 65. Cho tam gi\u00e1c ABC c\u00f3 tia ph\u00e2n gi\u00e1c c\u1ee7a g\u00f3c B v\u00e0 C c\u1eaft nhau t\u1ea1i D. \u0110\u01b0\u1eddng th\u1eb3ng qua A v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi AD c\u1eaft tia BD t\u1ea1i E, c\u1eaft CD t\u1ea1i F. Ch\u1ee9ng minh r\u1eb1ng: 1) T\u1ee9 gi\u00e1c AECD, AF DB n\u1ed9i ti\u1ebfp; 2) C\u00e1c \u0111\u01b0\u1eddng th\u1eb3ng AD, CE, BF \u0111\u1ed3ng quy. B\u00c0I TO\u00c1N CON B\u01af\u1edaM B\u00e0i 66. Cho \u0111\u01b0\u1eddng tr\u00f2n (O) v\u1edbi d\u00e2y cung AB. G\u1ecdi I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AB, qua I d\u1ef1ng hai d\u00e2y cung M N v\u00e0 P Q sao cho M P v\u00e0 N Q c\u1eaft AB l\u1ea7n l\u01b0\u1ee3t t\u1ea1i E v\u00e0 F. Ch\u1ee9ng minh r\u1eb1ng I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n th\u1eb3ng EF. B\u00e0i 67. Cho \u0111\u01b0\u1eddng tr\u00f2n (C) c\u00f3 M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a d\u00e2y cung P Q. G\u1ecdi AB, CD l\u00e0 hai d\u00e2y cung qua \u0111i\u1ec3m M. G\u1ecdi H, K l\u1ea7n l\u01b0\u1ee3t l\u00e0 giao \u0111i\u1ec3m c\u1ee7a P Q v\u1edbi AC v\u00e0 BD. Ch\u1ee9ng minh r\u1eb1ng HA.HC KB.KD =. HM2 KM2 B\u00e0i 68. Cho \u25b3ABC n\u1ed9i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n (O), I l\u00e0 t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n n\u1ed9i ti\u1ebfp. \u0110\u01b0\u1eddng th\u1eb3ng BI, 98 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 CI c\u1eaft \u0111\u01b0\u1eddng tr\u00f2n (O) t\u1ea1i E, F. G\u1ecdi K, D l\u1ea7n l\u01b0\u1ee3t l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AI v\u1edbi EF v\u00e0 BC. Bi\u1ebft AB + AC = 2BC. Ch\u1ee9ng minh r\u1eb1ng IK = ID. B\u00e0i 69. Cho hai \u0111\u01b0\u1eddng tr\u00f2n (C) v\u00e0 (C\u2032) c\u00f3 c\u00f9ng b\u00e1n k\u00ednh, c\u1eaft nhau t\u1ea1i hai \u0111i\u1ec3m A, B. G\u1ecdi O l\u00e0 trung \u0111i\u1ec3m AB. D\u00e2y cung CD c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n (C) qua \u0111i\u1ec3m O. G\u1ecdi P l\u00e0 giao \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n th\u1eb3ng CD c\u1eaft (C\u2032). EF l\u00e0 d\u00e2y cung (C\u2032) qua O v\u00e0 \u0111o\u1ea1n th\u1eb3ng EF c\u1eaft (C\u2032) t\u1ea1i Q. Ch\u1ee9ng minh r\u1eb1ng AB, CQ, EP \u0111\u1ed3ng quy. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 B\u00e0i 5 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M \u00d4n t\u1eadp ch\u01b0\u01a1ng 3 B\u00e0i 1. T\u1eeb \u0111i\u1ec3m A \u1edf ngo\u00e0i (O) , v\u1ebd hai ti\u1ebfp tuy\u1ebfn AM v\u00e0 AN t\u1ea1o v\u1edbi nhau M AN = 60\u25e6. T\u00ednh s\u1ed1 \u0111o cung l\u1edbn M N. B\u00e0i 2. Cho h\u00ecnh v\u1ebd: Bi\u1ebft AD l\u00e0 \u0111\u01b0\u1eddng k\u00ednh c\u1ee7a (O) , ABC = 70\u25e6. T\u00ednh s\u1ed1 \u0111o CAD. B\u00e0i 3. Tam gi\u00e1c ABC n\u1ed9i ti\u1ebfp (O) . Bi\u1ebft B = 32\u25e6, C = 40\u25e6. T\u00ednh s\u1ed1 \u0111o g\u00f3c \u1edf t\u00e2m BOC. B\u00e0i 4. Cho (O) , d\u00e2y AB, \u0111i\u1ec3m C thu\u1ed9c tia \u0111\u1ed1i c\u1ee7a tia AB. K\u1ebb ti\u1ebfp tuy\u1ebfn CD v\u1edbi \u0111\u01b0\u1eddng tr\u00f2n. Qua D k\u1ebb \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi tia ph\u00e2n gi\u00e1c c\u1ee7a ACD c\u1eaft \u0111\u01b0\u1eddng tr\u00f2n \u1edf E. Ch\u1ee9ng minh \u2322\u2322 r\u1eb1ng s\u0111AE = s\u0111BE. B\u00e0i 5. Cho tam gi\u00e1c ABC n\u1ed9i ti\u1ebfp (O) , ti\u1ebfp tuy\u1ebfn t\u1ea1i A c\u1eaft BC \u1edf D. Ch\u1ee9ng minh r\u1eb1ng DA2 = DB.DC. B\u00e0i 6. T\u1ee9 gi\u00e1c ABCD n\u1ed9i ti\u1ebfp \u0111\u01b0\u1eddng tr\u00f2n \u0111\u01b0\u1eddng k\u00ednh CD, bi\u1ebft DAB = 125\u25e6. T\u00ednh s\u1ed1 \u0111o cung nh\u1ecf BC. B\u00e0i 7. Cho (O) , AB v\u00e0 AC l\u00e0 hai ti\u1ebfp tuy\u1ebfn c\u1ee7a (O) t\u1ea1i B, C. Ch\u1ee9ng minh r\u1eb1ng T\u1ee9 gi\u00e1c AOBC n\u1ed9i ti\u1ebfp. | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 99","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m B\u00e0i 8. Cho g\u00f3c nh\u1ecdn xOy. Tr\u00ean c\u1ea1nh Ox, l\u1ea5y hai \u0111i\u1ec3m A v\u00e0 B sao cho OA = 2cm, OB = 6cm. Tr\u00ean c\u1ea1nh Oy l\u1ea5y hai \u0111i\u1ec3m C v\u00e0 D sao cho OC = 1, 5cm, OD = 8cm. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c ABDC n\u1ed9i ti\u1ebfp. B\u00e0i 9. Cho (O; 4) , d\u00e2y AB = AC; BAC = 45\u25e6. \u0110\u1ed9 d\u00e0i d\u00e2y BC l\u00e0 bao nhi\u00eau? B\u00e0i 10. Cho tam gi\u00e1c ABC n\u1ed9i ti\u1ebfp (O) v\u00e0 tia ph\u00e2n gi\u00e1c c\u1ee7a g\u00f3c A c\u1eaft \u0111\u01b0\u1eddng tr\u00f2n t\u1ea1i M, v\u1ebd \u0111\u01b0\u1eddng cao AH. Ch\u1ee9ng minh r\u1eb1ng: | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT 1) OM \u0111i qua trung \u0111i\u1ec3m c\u1ee7a d\u00e2y BC; 2) AM l\u00e0 tia ph\u00e2n gi\u00e1c c\u1ee7a AOH. B\u00e0i 11. Tam gi\u00e1c ABC c\u00e2n \u1edf A n\u1ed9i ti\u1ebfp (O) , ACB = 50\u25e6, D thu\u1ed9c cung nh\u1ecf BC sao cho BCD = 50\u25e6, AD c\u1eaft BC t\u1ea1i H. T\u00ednh s\u1ed1 \u0111o AHC. \u2322 B\u00e0i 12. AB ti\u1ebfp x\u00fac v\u1edbi (O) t\u1ea1i B, S l\u00e0 giao \u0111i\u1ec3m c\u1ee7a OA v\u1edbi (O) , bi\u1ebft s\u0111SB = 67\u25e6, s\u1ed1 \u0111o g\u00f3c AOB l\u00e0 bao nhi\u00eau? B\u00e0i 13. Cho \u0111\u01b0\u1eddng tr\u00f2n \u0111\u01b0\u1eddng k\u00ednh AB, l\u1ea5y C tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n sao cho CAB = 40\u25e6, D tr\u00ean n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n kh\u00f4ng ch\u1ee9a C sao cho BCD = 30\u25e6, AD c\u1eaft BC t\u1ea1i H. T\u00ednh AHC. B\u00e0i 14. Tam gi\u00e1c ABC n\u1ed9i ti\u1ebfp (O) , s\u1ed1 \u0111o cung nh\u1ecf AB l\u00e0 bao nhi\u00eau n\u1ebfu ACB = 40\u25e6? B\u00e0i 15. Cho (O) , \u0111\u01b0\u1eddng k\u00ednh AD; CB tr\u00ean (O) sao cho tia AD n\u1eb1m gi\u1eefa hai tia AC v\u00e0 AB, bi\u1ebft ACB = 40\u25e6. T\u00ednh DAB. B\u00e0i 16. Cho tam gi\u00e1c ABC n\u1ed9i ti\u1ebfp (O) , bi\u1ebft AOB = 100\u25e6, \u2322 = 90\u25e6. T\u00ednh \u2322 nh\u1ecf v\u00e0 s\u0111AC s\u0111BC l\u1edbn. B\u00e0i 17. Tam gi\u00e1c ABC c\u00e2n t\u1ea1i A c\u00f3 c\u1ea1nh \u0111\u00e1y nh\u1ecf h\u01a1n c\u1ea1nh b\u00ean, n\u1ed9i ti\u1ebfp (O) . Ti\u1ebfp tuy\u1ebfn t\u1ea1i B v\u00e0 C c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n l\u1ea7n l\u01b0\u1ee3t c\u1eaft tia AC v\u00e0 tia AB \u1edf E v\u00e0 D. Ch\u1ee9ng minh r\u1eb1ng: 1) BD2 = AD.CD; 2) T\u1ee9 gi\u00e1c BCDE n\u1ed9i ti\u1ebfp; 3) BC\/\/DE. \u221a a3 B\u00e0i 18. Cho tam gi\u00e1c ABC n\u1ed9i ti\u1ebfp (O) , AB = AC = a, \u0111\u01b0\u1eddng cao AH = . S\u1ed1 \u0111o cung 2 nh\u1ecf AB l\u00e0 bao nhi\u00eau? B\u00e0i 19. Cho (O) , M n\u1eb1m ngo\u00e0i (O) , v\u1ebd hai c\u00e1t tuy\u1ebfn M CA v\u00e0 M DB, N l\u00e0 giao \u0111i\u1ec3m c\u1ee7a AD v\u00e0 BC. Bi\u1ebft AN B = 40\u25e6, AM D = 20\u25e6. T\u00ednh s\u1ed1 \u0111o c\u00e1c cung nh\u1ecf AB v\u00e0 CD. B\u00e0i 20. Cho tam gi\u00e1c ABC, \u0111\u01b0\u1eddng cao AD, BE, CF c\u1eaft nhau t\u1ea1i H. T\u00ecm c\u00e1c t\u1ee9 gi\u00e1c n\u1ed9i ti\u1ebfp c\u00f3 trong h\u00ecnh. B\u00e0i 21. Tam gi\u00e1c ABC c\u00f3 AB = AC; d\/\/BC c\u1eaft AB, AC t\u1ea1i E v\u00e0 F. Ch\u1ee9ng minh r\u1eb1ng t\u1ee9 gi\u00e1c BEF C n\u1ed9i ti\u1ebfp. \u221a B\u00e0i 22. Trong (O; R) , d\u00e2y AB = R 2. \u0110\u1ed9 d\u00e0i cung nh\u1ecf AB l\u00e0 bao nhi\u00eau? 100 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 B\u00e0i 23. Cho (O) , \u0111\u01b0\u1eddng k\u00ednh AB, M l\u00e0 \u0111i\u1ec3m tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n, ti\u1ebfp tuy\u1ebfn t\u1ea1i B c\u1ee7a (O) c\u1eaft tia AM t\u1ea1i I. Ch\u1ee9ng minh r\u1eb1ng IBM = M AB. B\u00e0i 24. Cho (O) c\u00f3 d\u00e2y N P v\u00e0 QM c\u1eaft nhau t\u1ea1i F ngo\u00e0i (O) , N P v\u00e0 P Q c\u1eaft nhau t\u1ea1i E, F = 50\u25e6, \u2322 = 140\u25e6. Ch\u1ee9ng minh r\u1eb1ng M N \u22a5P Q. s\u0111N Q \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 101","Ch\u01b0\u01a1ng 4 H\u00ccNH TR\u1ee4 - H\u00ccNH N\u00d3N - H\u00ccNH C\u1ea6U B\u00e0i 1 H\u1ec7 th\u1ee9c h\u00ecnh tr\u00f2n | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M I. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n: C\u00e1c h\u1ec7 th\u1ee9c c\u1ee7a h\u00ecnh tr\u00f2n: Chu vi h\u00ecnh tr\u00f2n: C = 2\u03c0R \u0110\u1ed9 d\u00e0i cung tr\u00f2n c\u00f3 g\u00f3c \u1edf t\u00e2m n\u25e6: l = n.\u03c0 180\u25e6 Di\u1ec7n t\u00edch h\u00ecnh tr\u00f2n: S = \u03c0R2 Di\u1ec7n t\u00edch h\u00ecnh qu\u1ea1t c\u00f3 g\u00f3c \u1edf t\u00e2m n\u25e6: Squ\u1ea1t = n.\u03c0.R2 = l.R 360\u25e6 2 Di\u1ec7n t\u00edch h\u00ecnh vi\u00ean ph\u00e2n: Svp = Squ\u1ea1t \u2212 S\u25b3 Di\u1ec7n t\u00edch h\u00ecnh v\u00e0nh kh\u0103n: Svk = \u03c0. R2 \u2212 r2 v\u1edbi (R > r) II. B\u00e0i t\u1eadp: B\u00e0i 1. Cho (O; R) c\u00f3 di\u1ec7n t\u00edch 530, 66cm2. T\u00ednh R. B\u00e0i 2. H\u00ecnh tr\u00f2n c\u00f3 di\u1ec7n t\u00edch 12, 56cm2. T\u00ednh chu vi c\u1ee7a h\u00ecnh tr\u00f2n tr\u00ean. B\u00e0i 3. Cho h\u00ecnh tr\u00f2n c\u00f3 di\u1ec7n t\u00edch 234, 34cm2. T\u00ednh chu vi c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n n\u00f3i tr\u00ean. B\u00e0i 4. Cho \u0111\u01b0\u1eddng tr\u00f2n c\u00f3 chu vi l\u00e0 75, 36cm. T\u00ednh di\u1ec7n t\u00edch c\u1ee7a h\u00ecnh n\u00f3n n\u00f3i tr\u00ean. B\u00e0i 5. T\u00ednh di\u1ec7n t\u00edch m\u1ed9t h\u00ecnh tr\u00f2n bi\u1ebft chu vi c\u1ee7a n\u00f3 l\u00e0 9, 42cm. B\u00e0i 6. \u0110\u01b0\u1eddng tr\u00f2n c\u00f3 di\u1ec7n t\u00edch 0, 25\u03c0 th\u00ec chu vi l\u00e0 bao nhi\u00eau? B\u00e0i 7. Cho (O; 8cm) . T\u00ednh \u0111\u1ed9 d\u00e0i cung tr\u00f2n c\u00f3 s\u1ed1 \u0111o 120\u25e6 c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n tr\u00ean. B\u00e0i 8. Cho (O; R) bi\u1ebft \u0111\u1ed9 d\u00e0i cung tr\u00f2n c\u00f3 s\u1ed1 \u0111o 45\u25e6 l\u00e0 28, 62cm. T\u00ednh R. B\u00e0i 9. Cho (O; 9cm) . T\u00ednh di\u1ec7n t\u00edch h\u00ecnh qu\u1ea1t tr\u00f2n c\u00f3 s\u1ed1 \u0111o cung qu\u1ea1t 240\u25e6. B\u00e0i 10. Cho (O; R) bi\u1ebft di\u1ec7n t\u00edch h\u00ecnh qu\u1ea1t tr\u00f2n c\u00f3 s\u1ed1 \u0111o cung qu\u1ea1t 60\u25e6 l\u00e0 75, 36cm2. T\u00ednh R. B\u00e0i 11. Cho (O; 10cm) . T\u00ednh di\u1ec7n t\u00edch h\u00ecnh qu\u1ea1t tr\u00f2n c\u00f3 s\u1ed1 \u0111o cung qu\u1ea1t l\u00e0 120\u25e6. B\u00e0i 12. Cho (O; R) bi\u1ebft \u0111\u1ed9 d\u00e0i cung tr\u00f2n c\u00f3 s\u1ed1 \u0111o 135\u25e6 l\u00e0 85, 86cm. T\u00ednh R. | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 102","GI\u00c1O TR\u00ccNH TO\u00c1N 9 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M B\u00e0i 13. Cho h\u00ecnh tr\u00f2n c\u00f3 di\u1ec7n t\u00edch 706, 86cm2. T\u00ednh chu vi c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n n\u00f3i tr\u00ean. \u221a B\u00e0i 14. T\u00ednh \u0111\u1ed9 d\u00e0i (O; R) n\u1ed9i ti\u1ebfp h\u00ecnh vu\u00f4ng c\u00f3 c\u1ea1nh 3 2cm. B\u00e0i 15. T\u00ednh di\u1ec7n t\u00edch h\u00ecnh v\u00e0nh kh\u0103n \u0111\u01b0\u1ee3c t\u1ea1o b\u1edfi hai \u0111\u01b0\u1eddng tr\u00f2n (O; 5cm) v\u00e0 (O; 3cm) . B\u00e0i 16. Cho AB l\u00e0 cung nh\u1ecf c\u1ee7a (O; 10cm) , bi\u1ebft AOB = 60\u25e6. T\u00ednh \u0111\u1ed9 d\u00e0i cung l\u1edbn AB. B\u00e0i 17. T\u00ednh di\u1ec7n t\u00edch h\u00ecnh qu\u1ea1t tr\u00f2n 120\u25e6 c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n c\u00f3 b\u00e1n k\u00ednh 3cm. \u221a B\u00e0i 18. T\u00ednh di\u1ec7n t\u00edch h\u00ecnh v\u00e0nh kh\u0103n t\u1ea1o b\u1edfi hai \u0111\u01b0\u1eddng tr\u00f2n (O; 4cm)v\u00e0 O; 3 2cm . B\u00e0i 19. Cho (O; 2cm) ; A, B thu\u1ed9c (O) sao cho AOB = 75\u25e6. T\u00ednh di\u1ec7n t\u00edch h\u00ecnh qu\u1ea1t AOB. B\u00e0i 20. H\u00ecnh tr\u00f2n c\u00f3 di\u1ec7n t\u00edch l\u00e0 18\u03c0; b\u00e1n k\u00ednh c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n l\u00e0 bao nhi\u00eau? B\u00e0i 21. \u0110\u1ed9 d\u00e0i cung 60\u25e6 c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n c\u00f3 b\u00e1n k\u00ednh b\u1eb1ng 2dm l\u00e0 bao nhi\u00eau? B\u00e0i 22. M\u1ed9t cung tr\u00f2n c\u00f3 b\u00e1n k\u00ednh 3, \u0111\u1ed9 d\u00e0i \u03c0 c\u00f3 s\u1ed1 \u0111o l\u00e0 bao nhi\u00eau? B\u00e0i 23. M\u1ed9t cung tr\u00f2n c\u00f3 s\u1ed1 \u0111o 60\u25e6 c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n d\u00e0i 4\u03c0 th\u00ec b\u00e1n k\u00ednh l\u00e0 bao nhi\u00eau? 3 \u2322 B\u00e0i 24. Cho (O; R) v\u00e0 d\u00e2y AB, s\u0111AB = 90\u25e6. T\u00ednh chu vi h\u00ecnh vi\u00ean ph\u00e2n gi\u1edbi h\u1ea1n b\u1edfi d\u00e2y AB v\u00e0 cung AB. \u03c0R2 B\u00e0i 25. M\u1ed9t h\u00ecnh qu\u1ea1t c\u1ee7a (O; R) c\u00f3 di\u1ec7n t\u00edch . T\u00ednh \u0111\u1ed9 d\u00e0i cung t\u1ea1o th\u00e0nh. 3 B\u00e0i 26. T\u00ednh di\u1ec7n t\u00edch v\u00e0nh kh\u0103n gi\u1edbi h\u1ea1n b\u1edfi (O; 10) v\u00e0 (O; 6) . \u03c0 B\u00e0i 27. \u0110\u01b0\u1eddng tr\u00f2n c\u00f3 b\u00e1n k\u00ednh 6cm, m\u1ed9t cung c\u00f3 \u0111\u1ed9 d\u00e0i cm. T\u00ednh di\u1ec7n t\u00edch h\u00ecnh qu\u1ea1t \u0111\u01b0\u1ee3c 3 t\u1ea1o th\u00e0nh. \u221a B\u00e0i 28. T\u00ednh \u0111\u1ed9 d\u00e0i cung 90\u25e6 c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n b\u00e1n k\u00ednh 2. B\u00e0i 29. N\u1ebfu chu vi \u0111\u01b0\u1eddng tr\u00f2n t\u0103ng th\u00eam \u03c0cm th\u00ec b\u00e1n k\u00ednh c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n t\u0103ng th\u00eam bao nhi\u00eau? B\u00e0i 30. T\u00ednh b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c ABC \u0111\u1ec1u c\u00f3 c\u1ea1nh 5cm. B\u00e0i 31. T\u00ednh b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n n\u1ed9i ti\u1ebfp l\u1ee5c gi\u00e1c \u0111\u1ec1u c\u1ea1nh 2cm. B\u00e0i 32. Cho \u0111\u01b0\u1eddng tr\u00f2n (O; 3cm) v\u00e0 d\u00e2y cung AB = 3cm. T\u00ednh \u0111\u1ed9 d\u00e0i cung nh\u1ecf AB? B\u00e0i 33. M\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n c\u00f3 \u0111\u1ed9 d\u00e0i 6, 28cm, tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n l\u1ea5y hai \u0111i\u1ec3m A, B sao cho s\u0111AB = 50\u25e6. T\u00ednh \u0111\u1ed9 d\u00e0i cung AB n\u00e0y? B\u00e0i 34. T\u00ednh di\u1ec7n t\u00edch h\u00ecnh tr\u00f2n ngo\u1ea1i ti\u1ebfp h\u00ecnh vu\u00f4ng c\u1ea1nh 6cm. B\u00e0i 35. T\u00ednh di\u1ec7n t\u00edch h\u00ecnh qu\u1ea1t tr\u00f2n c\u00f3 b\u00e1n k\u00ednh 3cm v\u00e0 g\u00f3c \u1edf t\u00e2m t\u01b0\u01a1ng \u1ee9ng l\u00e0 50\u25e6. B\u00e0i 36. Cho \u0111\u01b0\u1eddng tr\u00f2n (O; R) v\u00e0 d\u00e2y cung AB = R. T\u00ednh di\u1ec7n t\u00edch h\u00ecnh vi\u00ean ph\u00e2n gi\u1edbi h\u1ea1n b\u1edfi d\u00e2y v\u00e0 cung nh\u1ecf AB? B\u00e0i 37. Cho hai h\u00ecnh tr\u00f2n (O; 6cm) v\u00e0 (O; 3cm) . T\u00ednh di\u1ec7n t\u00edch ph\u1ea7n gi\u1edbi h\u1ea1n ngo\u00e0i h\u00ecnh tr\u00f2n nh\u1ecf v\u00e0 trong h\u00ecnh tr\u00f2n l\u1edbn (h\u00ecnh v\u00e0nh kh\u0103n). B\u00e0i 38. T\u00ednh \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp l\u1ee5c gi\u00e1c \u0111\u1ec1u c\u1ea1nh 4cm. B\u00e0i 39. T\u00ednh t\u1ec9 s\u1ed1 b\u00e1n k\u00ednh c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n n\u1ed9i ti\u1ebfp v\u00e0 ngo\u1ea1i ti\u1ebfp h\u00ecnh tam gi\u00e1c \u0111\u1ec1u? | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 103","| Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m B\u00e0i 40. Cho tam gi\u00e1c ABC n\u1ed9i ti\u1ebfp (O; R) c\u00f3 C = 45\u25e6. T\u00ednh di\u1ec7n t\u00edch h\u00ecnh qu\u1ea1t tr\u00f2n AOB \u1ee9ng v\u1edbi cung nh\u1ecf AB? \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 B\u00e0i 2 H\u00ecnh tr\u1ee5 | C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT I. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n: Cho h\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y l\u00e0 R v\u00e0 chi\u1ec1u cao l\u00e0 h th\u00ec: Di\u1ec7n t\u00edch xung quanh: Sxq = 2\u03c0Rh Di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n: Stp = 2\u03c0Rh + 2\u03c0R2 Th\u1ec3 t\u00edch: V = \u03c0R2h II. B\u00e0i t\u1eadp: B\u00e0i 1. M\u1ed9t h\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y 5cm; cao 10cm. T\u00ednh: DTXQ; DTTP v\u00e0 TT c\u1ee7a h\u00ecnh tr\u1ee5. B\u00e0i 2. M\u1ed9t h\u00ecnh tr\u1ee5 c\u00f3 \u0111\u01b0\u1eddng k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y l\u00e0 6cm, chi\u1ec1u cao 10cm th\u00ec di\u1ec7n t\u00edch xung quanh b\u1eb1ng? B\u00e0i 3. T\u00ednh di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n v\u00e0 th\u1ec3 t\u00edch h\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y r = 3, 1cm v\u00e0 chi\u1ec1u cao h = 2, 4cm. B\u00e0i 4. M\u1ed9t h\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y 7cm, di\u1ec7n t\u00edch xung quanh 352cm2. T\u00ednh chi\u1ec1u cao v\u00e0 th\u1ec3 t\u00edch h\u00ecnh tr\u1ee5. B\u00e0i 5. M\u1ed9t h\u00ecnh tr\u1ee5 c\u00f3 chu vi \u0111\u00e1y b\u1eb1ng 20cm, di\u1ec7n t\u00edch xung quanh b\u1eb1ng 140cm2. T\u00ednh chi\u1ec1u cao c\u1ee7a h\u00ecnh tr\u1ee5. B\u00e0i 6. M\u1ed9t h\u00ecnh tr\u1ee5 c\u00f3 chi\u1ec1u cao b\u1eb1ng 8cm, di\u1ec7n t\u00edch xung quanh b\u1eb1ng 352cm2. T\u00ednh b\u00e1n k\u00ednh \u0111\u00e1y v\u00e0 th\u1ec3 t\u00edch c\u1ee7a h\u00ecnh tr\u1ee5. B\u00e0i 7. Th\u1ec3 t\u00edch h\u00ecnh tr\u1ee5 l\u00e0 942cm3. T\u00ednh di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n bi\u1ebft \u0111\u01b0\u1eddng cao l\u00e0 12cm. B\u00e0i 8. Cho h\u00ecnh ch\u1eef nh\u1eadt c\u00f3 chi\u1ec1u d\u00e0i 3cm, chi\u1ec1u r\u1ed9ng 2cm. Quay h\u00ecnh ch\u1eef nh\u1eadt m\u1ed9t v\u00f2ng quanh chi\u1ec1u d\u00e0i c\u1ee7a n\u00f3 \u0111\u01b0\u1ee3c m\u1ed9t h\u00ecnh tr\u1ee5. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 \u0111\u00f3 l\u00e0? B\u00e0i 9. M\u1ed9t h\u00ecnh tr\u1ee5 c\u00f3 di\u1ec7n t\u00edch xung quanh l\u00e0 128\u03c0cm2, chi\u1ec1u cao b\u1eb1ng b\u00e1n k\u00ednh \u0111\u00e1y. Khi \u0111\u00f3 th\u1ec3 t\u00edch c\u1ee7a n\u00f3 b\u1eb1ng? \u221a B\u00e0i 10. Cho h\u00ecnh ch\u1eef nh\u1eadt M N P Q c\u00f3 M N = 3N P ; N P = 5. T\u00ednh th\u1ec3 t\u00edch h\u00ecnh t\u1ea1o th\u00e0nh khi quay h\u00ecnh ch\u1eef nh\u1eadt M N P Q m\u1ed9t v\u00f2ng quanh M N. 104 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 B\u00e0i 11. Di\u1ec7n t\u00edch xung quanh h\u00ecnh tr\u1ee5 l\u00e0 452, 16cm2; chi\u1ec1u cao h\u00ecnh tr\u1ee5 l\u00e0 12cm, b\u00e1n k\u00ednh h\u00ecnh tr\u00f2n \u0111\u00e1y l\u00e0 bao nhi\u00eau? B\u00e0i 12. Cho h\u00ecnh ch\u1eef nh\u1eadt ABCD c\u00f3 AB = 5cm; BC = 12cm. T\u00ednh th\u1ec3 t\u00edch c\u1ee7a h\u00ecnh t\u1ea1o th\u00e0nh khi quay h\u00ecnh ch\u1eef nh\u1eadt ABCD quanh AD (\u03c0 \u2248 3, 14) . \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 B\u00e0i 3 H\u00ecnh n\u00f3n | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M I. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n: Cho h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y l\u00e0 R, chi\u1ec1u cao l\u00e0 h v\u00e0 \u0111\u01b0\u1eddng sinh l\u00e0 l th\u00ec: Di\u1ec7n t\u00edch xung quanh: Sxq = \u03c0Rl Di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n: Stp = \u03c0Rl + \u03c0R2 Th\u1ec3 t\u00edch: V = 1 \u03c0R2h 3 II. B\u00e0i t\u1eadp: B\u00e0i 1. M\u1ed9t h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y l\u00e0 12cm; chi\u1ec1u cao 16cm. 1) T\u00ednh \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh; 2) T\u00ednh di\u1ec7n t\u00edch xung quanh h\u00ecnh n\u00f3n. B\u00e0i 2. M\u1ed9t h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y l\u00e0 6cm, \u0111\u01b0\u1eddng sinh l\u00e0 8cm. T\u00ednh di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n c\u1ee7a h\u00ecnh n\u00f3n. B\u00e0i 3. M\u1ed9t h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y l\u00e0 3m, chi\u1ec1u cao l\u00e0 4m. T\u00ednh di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n. B\u00e0i 4. Khai tri\u1ec3n m\u1eb7t xung quanh c\u1ee7a m\u1ed9t h\u00ecnh n\u00f3n b\u00e1n k\u00ednh r, \u0111\u01b0\u1eddng sinh l\u00e0 l tr\u00ean m\u1eb7t qu\u1ea1t ta \u0111\u01b0\u1ee3c m\u1ed9t h\u00ecnh qu\u1ea1t. H\u00e3y x\u00e1c \u0111\u1ecbnh: B\u00e1n k\u00ednh h\u00ecnh qu\u1ea1t v\u00e0 \u0111\u1ed9 d\u00e0i cung h\u00ecnh qu\u1ea1t. B\u00e0i 5. Di\u1ec7n t\u00edch xung quanh m\u1ed9t h\u00ecnh n\u00f3n b\u1eb1ng 82\u03c0cm2. Di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n c\u1ee7a h\u00ecnh n\u00f3n b\u1eb1ng 16\u03c0cm2. T\u00ednh b\u00e1n k\u00ednh \u0111\u00e1y h\u00ecnh tr\u00f2n. B\u00e0i 6. M\u1ed9t h\u00ecnh n\u00f3n c\u00f3 \u0111\u01b0\u1eddng sinh l\u00e0 17cm. Bi\u1ebft r\u1eb1ng \u0111\u01b0\u1eddng cao l\u1edbn h\u01a1n b\u00e1n k\u00ednh l\u00e0 7cm. T\u00ednh th\u1ec3 t\u00edch c\u1ee7a h\u00ecnh n\u00f3n. B\u00e0i 7. M\u1ed9t h\u00ecnh n\u00f3n c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y 5cm; di\u1ec7n t\u00edch xung quanh 65\u03c0cm2. Th\u1ec3 t\u00edch h\u00ecnh n\u00f3n l\u00e0 bao nhi\u00eau? B\u00e0i 8. M\u1ed9t h\u00ecnh n\u00f3n c\u00f3 di\u1ec7n t\u00edch xung quanh 37, 68cm2; b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y l\u00e0 3cm. T\u00ednh | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 105","| C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT | Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng sinh. B\u00e0i 9. Cho tam gi\u00e1c ABC c\u00f3: A = 90\u25e6; AC = 18; AB = 24. T\u00ednh di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n c\u1ee7a h\u00ecnh n\u00f3n khi quay tam gi\u00e1c m\u1ed9t v\u00f2ng quanh AB. B\u00e0i 10. Cho tam gi\u00e1c ABC c\u00f3: A = 90\u25e6; BC = 2a; B = 30\u25e6. Quay tam gi\u00e1c quanh c\u1ea1nh AB ta \u0111\u01b0\u1ee3c h\u00ecnh n\u00f3n \u0111\u1ec9nh B. T\u00ednh di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n h\u00ecnh n\u00f3n. B\u00e0i 11. Th\u1ec3 t\u00edch m\u1ed9t h\u00ecnh n\u00f3n b\u1eb1ng 432\u03c0, chi\u1ec1u cao b\u1eb1ng 9. T\u00ednh di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n. B\u00e0i 12. Cho tam gi\u00e1c ABC vu\u00f4ng c\u00e2n t\u1ea1i A quay quanh c\u1ea1nh AB. T\u00ednh th\u1ec3 t\u00edch h\u00ecnh sinh ra \u221a b\u1edfi tam gi\u00e1c bi\u1ebft BC = 5 2cm. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 B\u00e0i 4 H\u00ecnh n\u00f3n c\u1ee5t I. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n: Cho h\u00ecnh n\u00f3n c\u1ee5t c\u00f3 chi\u1ec1u cao l\u00e0 h, \u0111\u01b0\u1eddng sinh l\u00e0 l, b\u00e1n k\u00ednh \u0111\u00e1y nh\u1ecf l\u00e0 r v\u00e0 b\u00e1n k\u00ednh \u0111\u00e1y l\u1edbn l\u00e0 R th\u00ec: Di\u1ec7n t\u00edch xung quanh: Sxq = \u03c0rl (R + r) Di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n: Stp = \u03c0 (R + r) + \u03c0R2 + \u03c0r2 Th\u1ec3 t\u00edch: V 1 R2 + r2 + R.r = \u03c0rh 3 II. B\u00e0i t\u1eadp: B\u00e0i 1. M\u1ed9t h\u00ecnh n\u00f3n c\u1ee5t c\u00f3 chi\u1ec1u cao 12cm; \u0111\u01b0\u1eddng sinh 20cm; b\u00e1n k\u00ednh \u0111\u00e1y l\u1edbn 24cm. T\u00ednh th\u1ec3 t\u00edch h\u00ecnh n\u00f3n c\u1ee5t. B\u00e0i 2. M\u1ed9t h\u00ecnh n\u00f3n c\u1ee5t c\u00f3 chi\u1ec1u cao 8cm; \u0111\u01b0\u1eddng sinh 10cm; b\u00e1n k\u00ednh \u0111\u00e1y l\u1edbn 12cm. T\u00ednh th\u1ec3 t\u00edch h\u00ecnh n\u00f3n c\u1ee5t. B\u00e0i 3. T\u00ednh th\u1ec3 t\u00edch c\u1ee7a h\u00ecnh n\u00f3n c\u1ee5t, b\u00e1n k\u00ednh hai \u0111\u00e1y b\u1eb1ng 2cm, 3cm, \u0111\u01b0\u1eddng cao b\u1eb1ng 6cm (Kh\u00f4ng y\u00eau c\u1ea7u v\u1ebd h\u00ecnh). B\u00e0i 4. M\u1ed9t h\u00ecnh n\u00f3n c\u1ee5t c\u00f3 chi\u1ec1u cao b\u1eb1ng 8; \u0111\u01b0\u1eddng sinh 10; b\u00e1n k\u00ednh \u0111\u00e1y l\u1edbn 12. T\u00ednh TT; DTXQ; DTTP c\u1ee7a h\u00ecnh n\u00f3n c\u1ee5t. B\u00e0i 5. M\u1ed9t h\u00ecnh n\u00f3n c\u1ee5t c\u00f3 chi\u1ec1u cao b\u1eb1ng 12; \u0111\u01b0\u1eddng sinh 13; b\u00e1n k\u00ednh \u0111\u00e1y l\u1edbn 8. T\u00ednh TT; 106 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 DTXQ; DTTP c\u1ee7a h\u00ecnh n\u00f3n c\u1ee5t. \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 B\u00e0i 5 H\u00ecnh c\u1ea7u I. Ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n: | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M Cho h\u00ecnh c\u1ea7u c\u00f3 b\u00e1n k\u00ednh \u0111\u00e1y l\u00e0 R th\u00ec: Di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u: S = 4\u03c0R2 Th\u1ec3 t\u00edch h\u00ecnh c\u1ea7u: V = 4 \u03c0R3 3 II. B\u00e0i t\u1eadp: B\u00e0i 1. M\u1ed9t qu\u1ea3 b\u00f3ng c\u00f3 \u0111\u01b0\u1eddng k\u00ednh 24cm. T\u00ednh di\u1ec7n t\u00edch b\u1ec1 m\u1eb7t v\u00e0 th\u1ec3 t\u00edch qu\u1ea3 b\u00f3ng. B\u00e0i 2. N\u1ebfu di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u l\u00e0 36\u03c0cm2. T\u00ednh b\u00e1n k\u00ednh v\u00e0 th\u1ec3 t\u00edch h\u00ecnh c\u1ea7u. B\u00e0i 3. H\u00ecnh c\u1ea7u c\u00f3 th\u1ec3 t\u00edch 7134, 56cm3. T\u00ednh b\u00e1n k\u00ednh h\u00ecnh c\u1ea7u. B\u00e0i 4. Bi\u1ebft di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u l\u00e0 400\u03c0cm2. T\u00ednh th\u1ec3 t\u00edch h\u00ecnh c\u1ea7u. B\u00e0i 5. Bi\u1ebft di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u l\u00e0 225\u03c0cm2. T\u00ednh th\u1ec3 t\u00edch h\u00ecnh c\u1ea7u. B\u00e0i 6. M\u1ed9t h\u00ecnh c\u1ea7u c\u00f3 s\u1ed1 \u0111o di\u1ec7n t\u00edch m2 g\u1ea5p 9 l\u1ea7n s\u1ed1 \u0111o th\u1ec3 t\u00edch m3 . T\u00ednh di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u. B\u00e0i 7. Di\u1ec7n t\u00edch xung quanh c\u1ee7a m\u1ed9t h\u00ecnh n\u00f3n b\u1eb1ng 135\u03c0cm2. Di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n c\u1ee7a n\u00f3 l\u00e0 216\u03c0cm2. Bi\u1ebft th\u1ec3 t\u00edch h\u00ecnh c\u1ea7u b\u1eb1ng 1 th\u1ec3 t\u00edch h\u00ecnh n\u00f3n. T\u00ednh th\u1ec3 t\u00edch h\u00ecnh c\u1ea7u? 3 B\u00e0i 8. C\u1eaft h\u00ecnh c\u1ea7u b\u1edfi m\u1ed9t m\u1eb7t ph\u1eb3ng \u0111i qua t\u00e2m, ta \u0111\u01b0\u1ee3c m\u1ed9t h\u00ecnh tr\u00f2n c\u00f3 chu vi b\u1eb1ng 20cm. T\u00ednh di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u. B\u00e0i 9. M\u1ed9t h\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y l\u00e0 5cm, chi\u1ec1u cao 6cm. M\u1ed9t h\u00ecnh c\u1ea7u c\u00f3 th\u1ec3 2 t\u00edch b\u1eb1ng th\u1ec3 t\u00edch h\u00ecnh tr\u1ee5 n\u00f3i tr\u00ean. T\u00ednh b\u00e1n k\u00ednh h\u00ecnh c\u1ea7u. 3 \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 B\u00e0i 6 \u00d4n t\u1eadp ch\u01b0\u01a1ng 4 B\u00e0i 1. Cho h\u00ecnh c\u1ea7u c\u00f3 th\u1ec3 t\u00edch 288\u03c0cm3. T\u00ednh di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u tr\u00ean. | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 107","| C\u01a1 s\u1edf d\u1ea1y th\u00eam & h\u1ecdc th\u00eam: L\u1edaP TO\u00c1N TT | Nguy\u1ec5n \u0110\u1ee9c Th\u1eafng - Ph\u1ea1m Ng\u1ecdc Tr\u00e2m B\u00e0i 2. Cho h\u00ecnh c\u1ea7u c\u00f3 s\u1ed1 \u0111o di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u b\u1eb1ng s\u1ed1 \u0111o th\u1ec3 t\u00edch. T\u00ednh b\u00e1n k\u00ednh c\u1ee7a h\u00ecnh c\u1ea7u. B\u00e0i 3. Cho h\u00ecnh n\u00f3n c\u00f3 chi\u1ec1u cao 12cm, b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y 5cm. T\u00ednh di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n v\u00e0 th\u1ec3 t\u00edch c\u1ee7a h\u00ecnh n\u00f3n tr\u00ean. B\u00e0i 4. Cho h\u00ecnh tr\u1ee5 c\u00f3 b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y l\u00e0 5cm, th\u1ec3 t\u00edch l\u00e0 100\u03c0cm3. T\u00ednh di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n c\u1ee7a h\u00ecnh tr\u1ee5. B\u00e0i 5. Cho h\u00ecnh n\u00f3n c\u1ee5t c\u00f3 b\u00e1n k\u00ednh \u0111\u01b0\u1eddng k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y l\u1edbn 7m, b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y nh\u1ecf 4m, chi\u1ec1u cao 4m. T\u00ednh di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n v\u00e0 th\u1ec3 t\u00edch c\u1ee7a h\u00ecnh n\u00f3n c\u1ee5t tr\u00ean. B\u00e0i 6. Cho h\u00ecnh tr\u1ee5 c\u00f3 chi\u1ec1u cao l\u00e0 12dm, di\u1ec7n t\u00edch c\u1ee7a h\u00ecnh tr\u00f2n \u0111\u00e1y l\u00e0 36\u03c0dm2. T\u00ednh di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5 tr\u00ean. B\u00e0i 7. Cho tam gi\u00e1c ABC vu\u00f4ng t\u1ea1i A c\u00f3 BC = 5cm, AC = 3cm. L\u1ea5y AB l\u00e0m tr\u1ee5c quay tam gi\u00e1c m\u1ed9t v\u00f2ng t\u1ea1o th\u00e0nh m\u1ed9t h\u00ecnh. H\u00e3y t\u00ednh di\u1ec7n t\u00edch xung quanh v\u00e0 th\u1ec3 t\u00edch c\u1ee7a h\u00ecnh \u0111\u00f3. B\u00e0i 8. Cho h\u00ecnh tr\u1ee5 c\u00f3 s\u1ed1 \u0111o th\u1ec3 t\u00edch g\u1ea5p \u0111\u00f4i di\u1ec7n t\u00edch xung quanh. T\u00ednh di\u1ec7n t\u00edch \u0111\u00e1y h\u00ecnh tr\u1ee5. B\u00e0i 9. Cho h\u00ecnh n\u00f3n c\u00f3 \u0111\u01b0\u1eddng sinh t\u1ea1o v\u1edbi tr\u1ee5c m\u1ed9t g\u00f3c 60\u25e6 bi\u1ebft \u0111\u01b0\u1eddng sinh 12cm. T\u00ednh th\u1ec3 t\u00edch h\u00ecnh n\u00f3n. B\u00e0i 10. Cho h\u00ecnh ch\u1eef nh\u1eadt ABCD c\u00f3 AC = 25cm, BC = 15cm. L\u1ea5y AB l\u00e0m tr\u1ee5c quay h\u00ecnh ch\u1eef nh\u1eadt m\u1ed9t v\u00f2ng t\u1ea1o th\u00e0nh m\u1ed9t h\u00ecnh. T\u00ednh di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n v\u00e0 th\u1ec3 t\u00edch c\u1ee7a h\u00ecnh v\u1eeba t\u1ea1o \u1edf tr\u00ean. B\u00e0i 11. Di\u1ec7n t\u00edch xung quanh m\u1ed9t h\u00ecnh tr\u1ee5 l\u00e0 96\u03c0cm2. Chi\u1ec1u cao h\u00ecnh tr\u1ee5 n\u00e0y l\u00e0 12cm. T\u00ednh b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y? B\u00e0i 12. Th\u1ec3 t\u00edch m\u1ed9t h\u00ecnh tr\u1ee5 l\u00e0 375\u03c0cm3. Chi\u1ec1u cao c\u1ee7a h\u00ecnh tr\u1ee5 l\u00e0 15cm. T\u00ednh di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh tr\u1ee5? B\u00e0i 13. Th\u1ec3 t\u00edch m\u1ed9t h\u00ecnh n\u00f3n b\u1eb1ng 432\u03c0cm3. Chi\u1ec1u cao c\u1ee7a h\u00ecnh n\u00f3n n\u00e0y l\u00e0 9cm. T\u00ecm di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n? B\u00e0i 14. Di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh n\u00f3n b\u1eb1ng 100\u03c0cm2, di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n c\u1ee7a n\u00f3 l\u00e0 136\u03c0cm2. T\u00ednh b\u00e1n k\u00ednh \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y c\u1ee7a h\u00ecnh n\u00f3n n\u00e0y? B\u00e0i 15. M\u1ed9t h\u00ecnh c\u1ea7u c\u00f3 b\u00e1n k\u00ednh 5cm. T\u00ecm di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u v\u00e0 th\u1ec3 t\u00edch h\u00ecnh c\u1ea7u \u0111\u00f3? B\u00e0i 16. Th\u1ec3 t\u00edch m\u1ed9t h\u00ecnh c\u1ea7u l\u00e0 972\u03c0cm3. T\u00ednh di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u \u0111\u00f3? B\u00e0i 17. Di\u1ec7n t\u00edch m\u1ed9t m\u1eb7t c\u1ea7u l\u00e0 \u03c0 cm2. T\u00ecm \u0111\u01b0\u1eddng k\u00ednh c\u1ee7a h\u00ecnh c\u1ea7u n\u00e0y? 4 B\u00e0i 18. Di\u1ec7n t\u00edch m\u1ed9t m\u1eb7t c\u1ea7u l\u00e0 9\u03c0cm2. T\u00ednh th\u1ec3 t\u00edch h\u00ecnh c\u1ea7u n\u00e0y? B\u00e0i 19. H\u00ecnh khai tri\u1ec3n c\u1ee7a m\u1eb7t xung quanh c\u1ee7a m\u1ed9t h\u00ecnh n\u00f3n l\u00e0 m\u1ed9t h\u00ecnh qu\u1ea1t. B\u00e1n k\u00ednh h\u00ecnh qu\u1ea1t \u0111\u00f3 l\u00e0 16cm, s\u1ed1 \u0111o cung l\u00e0 120\u25e6. Tan c\u1ee7a n\u1eefa g\u00f3c \u1edf \u0111\u1ec9nh h\u00ecnh n\u00f3n l\u00e0 bao nhi\u00eau? B\u00e0i 20. V\u1edbi m\u1ed9t c\u00e1i th\u01b0\u1edbc d\u00e2y, li\u1ec7u c\u00f3 th\u1ec3 x\u00e1c \u0111\u1ecbnh \u0111\u01b0\u1ee3c di\u1ec7n t\u00edch m\u1eb7t c\u1ea7u v\u00e0 th\u1ec3 t\u00edch c\u1ee7a m\u1ed9t v\u1eadt h\u00ecnh c\u1ea7u \u0111\u01b0\u1ee3c kh\u00f4ng? N\u00eau c\u00e1ch l\u00e0m? 108 N\u0103m h\u1ecdc: 2023 - 2024","GI\u00c1O TR\u00ccNH TO\u00c1N 9 | NGUY\u1ec4N \u0110\u1ee8C TH\u1eaeNG - PH\u1ea0M NG\u1eccC TR\u00c2M B\u00e0i 21. V\u1edbi hai qu\u1ea3 d\u01b0a h\u1ea5u (xem nh\u01b0 hai h\u00ecnh c\u1ea7u), m\u1ed9t to v\u00e0 m\u1ed9t nh\u1ecf, t\u1ec9 s\u1ed1 c\u00e1c \u0111\u01b0\u1eddng k\u00ednh c\u1ee7a ch\u00fang l\u00e0 5 : 4 nh\u01b0ng gi\u00e1 qu\u1ea3 to g\u1ea5p r\u01b0\u1ee1i gi\u00e1 qu\u1ea3 nh\u1ecf. B\u1ea1n ch\u1ecdn mua qu\u1ea3 n\u00e0o th\u00ec l\u1ee3i h\u01a1n? (Xem ch\u1ea5t l\u01b0\u1ee3ng c\u1ee7a ch\u00fang l\u00e0 nh\u01b0 nhau) B\u00e0i 22. M\u1ed9t m\u00e1ng h\u1ee9ng n\u01b0\u1edbc c\u00f3 d\u1ea1ng n\u1eeda h\u00ecnh tr\u1ee5, b\u1ecbt k\u00edn \u1edf m\u1ed9t \u0111\u1ea7u. \u0110\u01b0\u1eddng k\u00ednh l\u00f2ng m\u00e1ng 18cm, chi\u1ec1u d\u00e0i 6m. H\u1ecfi di\u1ec7n t\u00edch t\u1ea5m kim lo\u1ea1i k\u1ebdm \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng \u0111\u1ec3 ch\u1ebf t\u1ea1o c\u00e1i m\u00e1ng \u0111\u00f3? B\u00e0i 23. Ng\u01b0\u1eddi ta \u0111\u00e0o m\u1ed9t c\u00e1i gi\u1ebfng h\u00ecnh tr\u1ee5 c\u00f3 chi\u1ec1u s\u00e2u l\u00e0 12m v\u00e0 \u0111\u01b0\u1eddng k\u00ednh mi\u1ec7ng gi\u1ebfng l\u00e0 1, 50m. 1) T\u00ednh kh\u1ed1i l\u01b0\u1ee3ng \u0111\u1ea5t \u0111\u00e3 \u0111\u00e0o l\u00ean?; 2) Bi\u1ebft r\u1eb1ng \u0111\u1ea5t \u0111\u00e0o l\u00ean \u0111\u00e3 t\u01a1i th\u00eam 20% th\u1ec3 t\u00edch ban \u0111\u1ea7u. Ng\u01b0\u1eddi ta d\u00f9ng xe t\u1ea3i c\u00f3 kh\u1ea3 n\u0103ng chuy\u00ean ch\u1edf 5, 2m3 m\u1ed9t chuy\u1ebfn. H\u1ecfi ph\u1ea3i ch\u1edf bao nhi\u00eau chuy\u1ebfn xe m\u1edbi h\u1ebft ch\u1ed7 \u0111\u1ea5t \u0111\u00f3?; 3) Sau m\u1ed9t tr\u1eadn m\u01b0a, m\u1eb7t n\u01b0\u1edbc c\u00f2n c\u00e1ch mi\u1ec7ng gi\u1ebfng 8, 75m. H\u1ecfi th\u1ec3 t\u00edch n\u01b0\u1edbc trong gi\u1ebfng l\u00e0 bao nhi\u00eau m\u00e9t kh\u1ed1i? B\u00e0i 24. T\u1eeb m\u1ed9t kh\u1ed1i g\u1ed7 \u0111\u1eb7c h\u00ecnh tr\u1ee5 b\u00e1n k\u00ednh 12cm, chi\u1ec1u cao 25cm, ng\u01b0\u1eddi ta kho\u00e9t \u0111i m\u1ed9t c\u00e1i l\u1ed7 h\u00ecnh tr\u1ee5 c\u00f3 c\u00f9ng tr\u1ee5c v\u1edbi kh\u1ed1i g\u1ed7 ban \u0111\u1ea7u, b\u00e1n k\u00ednh 6cm. H\u1ecfi th\u1ec3 t\u00edch kh\u1ed1i g\u1ed7 c\u00f2n l\u1ea1i b\u1eb1ng bao nhi\u00eau th\u1ec3 t\u00edch kh\u1ed1i g\u1ed7 ban \u0111\u1ea7u? B\u00e0i 25. C\u00f3 th\u1ec3 x\u1ebfp \u0111\u01b0\u1ee3c bao nhi\u00eau qu\u1ea3 c\u1ea7u c\u00f3 b\u00e1n k\u00ednh 1cm v\u00e0o m\u1ed9t h\u00ecnh h\u1ed9p c\u00f3 \u0111\u00e1y h\u00ecnh vu\u00f4ng c\u1ea1nh 30cm, chi\u1ec1u cao 2cm? \u2212\u2212\u2212\u2212\u2212\u22c6\u22c6\u22c6\u2212\u2212\u2212\u2212\u2212 | L\u1edaP TO\u00c1N TT - 35F1 Chi L\u0103ng, P.9, TP. \u0110\u00e0 L\u1ea1t 109"]