Ghi ch\u00fa: T\u00e0i li\u1ec7u tr\u00edch d\u1eabn<\/p>\n\n\n\n
\u0110\u01b0\u1eddng tr\u00f2n l\u00e0 m\u1ed9t trong nh\u1eefng h\u00ecnh h\u1ecdc kh\u00f3 v\u1edbi nhi\u1ec1u d\u1ea1ng b\u00e0i t\u1eadp kh\u00e1c nhau. \u0110\u1ec3 c\u00f3 th\u1ec3 hi\u1ec3u \u0111\u01b0\u1ee3c \u0111\u1ea7y \u0111\u1ee7 v\u1ec1 h\u00ecnh h\u1ecdc n\u00e0y c\u1ea7n x\u00e1c \u0111\u1ecbnh \u0111\u01b0\u1ee3c l\u00fd thuy\u1ebft c\u0169ng nh\u01b0 b\u00e0i t\u1eadp v\u1ec1 \u0111\u01b0\u1eddng k\u00ednh v\u00e0 d\u00e2y c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n. Ch\u00ednh v\u00ec v\u1eady h\u00e3y c\u00f9ng Vietlearn.org\/ theo d\u00f5i b\u00e0i gi\u1ea3ng sau v\u1edbi nh\u1eefng b\u00e0i t\u1eadp th\u00f4ng d\u1ee5ng v\u1ec1 m\u1ea3ng ki\u1ebfn th\u1ee9c n\u00e0y \u0111\u1ec3 c\u00e1c b\u1ea1n h\u1ecdc sinh<\/a> c\u00f3 th\u1ec3 \u00e1p d\u1ee5ng.<\/p>\n\n\n\n I. L\u00fd thuy\u1ebft v\u1ec1 \u0111\u01b0\u1eddng k\u00ednh v\u00e0 d\u00e2y c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n:<\/p>\n\n\n\n Cho \u0111\u01b0\u1eddng tr\u00f2n c\u00f3 t\u00e2m l\u00e0 O v\u00e0 c\u00f3 2 \u0111i\u1ec3m ph\u00e2n bi\u1ec7t l\u00e0 A v\u00e0 B n\u1eb1m tr\u00ean \u0111\u01b0\u1eddng tr\u00f2n th\u00ec 2 \u0111i\u1ec3m n\u00e0y s\u1ebd chia \u0111\u01b0\u1eddng tr\u00f2n l\u00e0m 2 ph\u1ea7n. \u0110\u1ed1i v\u1edbi 2 ph\u1ea7n n\u00e0y, m\u1ed7i ph\u1ea7n s\u1ebd \u0111\u01b0\u1ee3c xem l\u00e0 m\u1ed9t cung.<\/p>\n\n\n\n Trong \u0111\u00f3:<\/p>\n\n\n\n 2 \u0111i\u1ec3m A v\u00e0 B \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 2 m\u00fat c\u1ee7a cung.<\/p>\n\n\n\n \u0110o\u1ea1n th\u1eb3ng n\u1ed1i li\u1ec1n 2 m\u00fat c\u1ee7a cung \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 d\u00e2y cung (d\u00e2y).<\/p>\n\n\n\n D\u00e2y cung \u0111i qua t\u00e2m c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 \u0111\u01b0\u1eddng k\u00ednh.<\/p>\n\n\n\n Ngo\u00e0i ra trong m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n c\u00f2n c\u00f3 m\u1ed1i quan h\u1ec7 gi\u1eefa \u0111\u01b0\u1eddng k\u00ednh v\u00e0 d\u00e2y cung nh\u01b0 sau: \u0110\u01b0\u1eddng k\u00ednh s\u1ebd d\u00e0i g\u1ea5p \u0111\u00f4i b\u00e1n k\u00ednh.<\/p>\n\n\n\n \u0110\u1ecbnh l\u00fd 1:<\/p>\n\n\n\n Trong t\u1ea5t c\u1ea3 c\u00e1c d\u00e2y cung c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n, d\u00e2y cung l\u1edbn nh\u1ea5t l\u00e0 \u0111\u01b0\u1eddng k\u00ednh.<\/p>\n\n\n\n X\u00e9t m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O, b\u00e1n k\u00ednh R, (O , R) ta c\u00f3:<\/p>\n\n\n\n A \u2208 O, B \u2208 O \u21d2 AB \u2264 2R<\/p>\n\n\n\n So s\u00e1nh gi\u1eefa \u0111\u1ed9 d\u00e0i d\u00e2y cung v\u00e0 \u0111\u01b0\u1eddng k\u00ednh c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O<\/p>\n\n\n\n Ch\u1ee9ng minh:<\/p>\n\n\n\n Tr\u01b0\u1eddng h\u1ee3p 1:<\/p>\n\n\n\n N\u1ebfu \u0111\u01b0\u1eddng k\u00ednh c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n l\u00e0 d\u00e2y AB th\u00ec AB = 2R<\/p>\n\n\n\n Tr\u01b0\u1eddng h\u1ee3p 2:<\/p>\n\n\n\n N\u1ebfu \u0111\u01b0\u1eddng k\u00ednh c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n kh\u00f4ng ph\u1ea3i l\u00e0 d\u00e2y AB.<\/p>\n\n\n\n X\u00e9t \u0394OAB, c\u00f3: AB < OA + OB = R + R = 2R<\/p>\n\n\n\n \u21d2 V\u1eady trong m\u1ed9t \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O, b\u00e1n k\u00ednh R, (O , R) ta lu\u00f4n c\u00f3 AB \u2264 2R<\/p>\n\n\n\n \u0110\u1ecbnh l\u00fd 2:<\/p>\n\n\n\n Quan h\u1ec7 vu\u00f4ng g\u00f3c gi\u1eefa \u0111\u01b0\u1eddng k\u00ednh c\u00f9ng v\u1edbi d\u00e2y \u0111\u01b0\u1eddng tr\u00f2n \u0111\u01b0\u1ee3c \u0111\u1ecbnh ngh\u0129a nh\u01b0 sau: N\u1ebfu \u0111\u01b0\u1eddng k\u00ednh m\u00e0 vu\u00f4ng g\u00f3c v\u1edbi 1 d\u00e2y trong 1 \u0111\u01b0\u1eddng tr\u00f2n th\u00ec \u0111i qua trung \u0111i\u1ec3m c\u1ee7a d\u00e2y \u0111\u00f3.<\/p>\n\n\n\n Ch\u1ee9ng minh:<\/p>\n\n\n\n Tr\u01b0\u1eddng h\u1ee3p 1:<\/p>\n\n\n\n N\u1ebfu \u0111\u01b0\u1eddng k\u00ednh c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n l\u00e0 CD th\u00ec hi\u1ec3n nhi\u00ean AB s\u1ebd \u0111i qua trung \u0111i\u1ec3m O (t\u00e2m) c\u1ee7a \u0111\u01b0\u1eddng k\u00ednh CD.<\/p>\n\n\n\n Tr\u01b0\u1eddng h\u1ee3p 2:<\/p>\n\n\n\n N\u1ebfu \u0111\u01b0\u1eddng k\u00ednh c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n kh\u00f4ng ph\u1ea3i l\u00e0 CD<\/p>\n\n\n\n Ta g\u1ecdi I l\u00e0 giao \u0111i\u1ec3m gi\u1eefa AB v\u00e0 CD. Khi \u0111\u00f3 \u0394OCD c\u00f3 OD = OC = R<\/p>\n\n\n\n \u21d2 \u0394OCD l\u00e0 tam gi\u00e1c c\u00e2n t\u1ea1i O (Do OC = OD = R)<\/p>\n\n\n\n \u21d2 \u0110\u01b0\u1eddng cao c\u1ee7a \u0394OCD l\u00e0 OI v\u00e0 c\u0169ng l\u00e0 \u0111\u01b0\u1eddng trung tuy\u1ebfn<\/p>\n\n\n\n \u21d2 ID = IC (Hay I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a CD)<\/p>\n\n\n\n Quan h\u1ec7 vu\u00f4ng g\u00f3c gi\u1eefa \u0111\u01b0\u1eddng k\u00ednh v\u00e0 d\u00e2y c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n<\/p>\n\n\n\n V\u1eady n\u1ebfu \u0111\u01b0\u1eddng k\u00ednh m\u00e0 vu\u00f4ng g\u00f3c v\u1edbi 1 d\u00e2y trong 1 \u0111\u01b0\u1eddng tr\u00f2n th\u00ec \u0111i qua trung \u0111i\u1ec3m c\u1ee7a d\u00e2y \u0111\u00f3. \u0110\u00e2y c\u0169ng ch\u00ednh l\u00e0 \u0111\u1ecbnh l\u00fd quan tr\u1ecdng trong to\u00e1n 9 \u0111\u01b0\u1eddng k\u00ednh v\u00e0 d\u00e2y c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n c\u1ea7n n\u1eafm v\u1eefng.<\/p>\n\n\n\n \u0110\u1ecbnh l\u00fd 3:<\/p>\n\n\n\n N\u1ebfu \u0111\u01b0\u1eddng k\u00ednh m\u00e0 \u0111i qua trung \u0111i\u1ec3m c\u1ee7a 1 d\u00e2y v\u00e0 kh\u00f4ng \u0111i qua t\u00e2m trong 1 \u0111\u01b0\u1eddng tr\u00f2n th\u00ec vu\u00f4ng g\u00f3c v\u1edbi d\u00e2y \u0111\u00f3.<\/p>\n\n\n\n Ch\u1ee9ng minh:<\/p>\n\n\n\n G\u1ecdi I l\u00e0 giao \u0111i\u1ec3m c\u1ee7a \u0111\u01b0\u1eddng k\u00ednh AB v\u00e0 d\u00e2y CD.<\/p>\n\n\n\n \u21d2 \u0394OCD l\u00e0 tam gi\u00e1c c\u00e2n t\u1ea1i O (Do OC = OD = R)<\/p>\n\n\n\n M\u00e0 ta c\u00f3 OI l\u00e0 trung tuy\u1ebfn c\u1ee7a \u0394OCD n\u00ean \u0111\u1ed3ng th\u1eddi c\u0169ng l\u00e0 \u0111\u01b0\u1eddng cao c\u1ee7a tam gi\u00e1c.<\/p>\n\n\n\n \u21d2 V\u1eady OI \u22a5 CD t\u1ea1i \u0111i\u1ec3m I.<\/p>\n\n\n\n \u0110\u01b0\u1eddng k\u00ednh qua trung \u0111i\u1ec3m 1 d\u00e2y kh\u00f4ng \u0111i qua t\u00e2m s\u1ebd vu\u00f4ng g\u00f3c v\u1edbi d\u00e2y \u0111\u00f3<\/p>\n\n\n\n L\u01b0u \u00fd:<\/p>\n\n\n\n Trong quan h\u1ec7 vu\u00f4ng g\u00f3c gi\u1eefa \u0111\u01b0\u1eddng k\u00ednh v\u1edbi d\u00e2y \u0111\u01b0\u1eddng tr\u00f2n, n\u1ebfu \u0111\u01b0\u1eddng k\u00ednh m\u00e0 \u0111i qua trung \u0111i\u1ec3m c\u1ee7a 1 d\u00e2y th\u00ec c\u00f3 th\u1ec3 s\u1ebd kh\u00f4ng vu\u00f4ng g\u00f3c v\u1edbi \u0111\u00f3.<\/p>\n\n\n\n Gi\u1ea3 s\u1eed 2 \u0111\u01b0\u1eddng k\u00ednh c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n (O , R) l\u00e0 AB v\u00e0 CD.<\/p>\n\n\n\n Khi \u0111\u00f3 ta c\u00f3 CD c\u0169ng l\u00e0 1 d\u00e2y cung c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O.<\/p>\n\n\n\n M\u00e0 O \u2208 CD \u0111\u1ed3ng th\u1eddi OC = OD (do CD l\u00e0 \u0111\u01b0\u1eddng k\u00ednh c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n t\u00e2m O)<\/p>\n\n\n\n \u21d2 O ch\u00ednh l\u00e0 trung \u0111i\u1ec3m c\u1ee7a CD.<\/p>\n\n\n\n \u21d2 Khi \u0111\u00f3, \u0111\u01b0\u1eddng k\u00ednh AB s\u1ebd \u0111i qua trung \u0111i\u1ec3m O c\u1ee7a CD nh\u01b0ng \u0111\u01b0\u1eddng k\u00ednh s\u1ebd kh\u00f4ng vu\u00f4ng g\u00f3c v\u1edbi d\u00e2y \u0111\u01b0\u1eddng tr\u00f2n.<\/p>\n\n\n\n L\u01b0u \u00fd trong quan h\u1ec7 vu\u00f4ng g\u00f3c gi\u1eefa \u0111\u01b0\u1eddng k\u00ednh v\u00e0 d\u00e2y \u0111\u01b0\u1eddng tr\u00f2n<\/p>\n\n\n\n S\u1ef1 x\u00e1c \u0111\u1ecbnh \u0111\u01b0\u1eddng tr\u00f2n t\u00ednh ch\u1ea5t \u0111\u1ed1i x\u1ee9ng c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n<\/p><\/blockquote>\n\n\n\n II. B\u00e0i t\u1eadp \u0111\u01b0\u1eddng k\u00ednh v\u00e0 d\u00e2y c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n sbt:<\/p>\n\n\n\n Tr\u00ean \u0111\u00e2y l\u00e0 b\u00e0i gi\u1ea3ng v\u1ec1 \u0111\u01b0\u1eddng k\u00ednh v\u00e0 d\u00e2y c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n, \u0111\u1ec3 c\u1ee7ng c\u1ed1 l\u1ea1i l\u00fd thuy\u1ebft c\u1ee7a b\u00e0i h\u1ecdc, d\u01b0\u1edbi \u0111\u00e2y l\u00e0 m\u1ed9t s\u1ed1 b\u00e0i to\u00e1n c\u01a1 b\u1ea3n v\u1ec1 d\u1ea1ng ki\u1ebfn th\u1ee9c n\u00e0y gi\u00fap c\u00e1c em h\u1ec7 th\u1ed1ng h\u00f3a l\u1ea1i ki\u1ebfn th\u1ee9c cho qu\u00e1 tr\u00ecnh \u00f4n t\u1eadp c\u0169ng nh\u01b0 luy\u1ec7n thi<\/a> \u0111\u01b0\u1ee3c hi\u1ec7u qu\u1ea3.<\/p>\n\n\n\n B\u00e0i t\u1eadp 1: (B\u00e0i 15\/SBT T\u1eadp 1, To\u00e1n 9, trang 158)<\/p>\n\n\n\n Cho \u0394ABC, \u0111\u01b0\u1eddng cao CK v\u00e0 BH. Ch\u1ee9ng minh r\u1eb1ng:<\/p>\n\n\n\n B, C, H, K l\u00e0 4 \u0111i\u1ec3m kh\u00f4ng c\u00f9ng thu\u1ed9c 1 \u0111\u01b0\u1eddng tr\u00f2n.<\/p>\n\n\n\n HK < BC.<\/p>\n\n\n\n B\u00e0i t\u1eadp 2: (B\u00e0i 16\/SBT T\u1eadp 1, To\u00e1n 9, trang 159)<\/p>\n\n\n\n Cho t\u1ee9 gi\u00e1c ABCD c\u00f3 g\u00f3c B = g\u00f3c B = 90\u2070<\/p>\n\n\n\n Ch\u1ee9ng minh A, B, C, D l\u00e0 4 \u0111i\u1ec3m c\u00f9ng thu\u1ed9c tr\u00ean 1 \u0111\u01b0\u1eddng tr\u00f2n.<\/p>\n\n\n\n So s\u00e1nh \u0111\u1ed9 d\u00e0i c\u1ee7a cung AB v\u00e0 cung BD. N\u1ebfu nh\u01b0 AC = BD th\u00ec ABCD l\u00e0 t\u1ee9 gi\u00e1c g\u00ec? T\u1ea1i sao?<\/p>\n\n\n\n B\u00e0i t\u1eadp 3: (B\u00e0i 17\/SBT T\u1eadp 1, To\u00e1n 9, trang 159)<\/p>\n\n\n\n Cho n\u1eeda \u0111\u01b0\u1eddng tr\u00f2n (O,AB) (T\u00e2m O, \u0111\u01b0\u1eddng k\u00ednh AB) v\u00e0 d\u00e2y EF kh\u00f4ng c\u1eaft v\u1edbi \u0111\u01b0\u1eddng k\u00ednh. G\u1ecdi K v\u00e0 I l\u1ea7n l\u01b0\u1ee3t l\u00e0 ch\u00e2n \u0111\u01b0\u1eddng vu\u00f4ng g\u00f3c k\u1ebb t\u1eeb B v\u00e0 A \u0111\u1ebfn d\u00e2y EF. Ch\u1ee9ng minh IE = KF.<\/p>\n\n\n\n B\u00e0i gi\u1ea3ng \u0111\u01b0\u1eddng k\u00ednh v\u00e0 d\u00e2y c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n tr\u00ean \u0111\u00e3 cung c\u1ea5p ki\u1ebfn th\u1ee9c v\u1ec1 l\u00fd thuy\u1ebft c\u0169ng nh\u01b0 c\u00e1c d\u1ea1ng b\u00e0i t\u1eadp c\u01a1 b\u1ea3n v\u1ec1 b\u00e0i h\u1ecdc n\u00e0y. Hy v\u1ecdng \u0111\u00e2y s\u1ebd l\u00e0 nh\u1eefng th\u00f4ng tin b\u1ed5 \u00edch d\u00e0nh cho c\u00e1c b\u1ea1n h\u1ecdc sinh<\/a> v\u00e0 qu\u00fd ph\u1ee5 huynh c\u00f3 nhu c\u1ea7u gi\u1ea3ng d\u1ea1y v\u00e0 \u00f4n t\u1eadp cho con em c\u1ee7a m\u00ecnh. \u0110\u1eebng qu\u00ean th\u01b0\u1eddng xuy\u00ean truy c\u1eadp Vietlearn.org\/ \u0111\u1ec3 c\u1eadp nh\u1eadt th\u00eam nh\u1eefng ki\u1ebfn th\u1ee9c m\u00f4n h\u1ecdc kh\u00e1c nh\u00e9!<\/p>\n\n\n\n T\u00ecm hi\u1ec3u th\u00eam:<\/p>\n\n\n\n Li\u00ean h\u1ec7 gi\u1eefa d\u00e2y v\u00e0 kho\u1ea3ng c\u00e1ch t\u1eeb t\u00e2m \u0111\u1ebfn d\u00e2y<\/p>\n\n\n\n Li\u00ean h\u1ec7 gi\u1eefa cung v\u00e0 d\u00e2y \u2013 L\u00fd thuy\u1ebft c\u1ea7n nh\u1edb c\u00f9ng Vietlearn<\/p>\n\n\n\n V\u1ecb t<\/p>\n\n\n\n\n\n\n","protected":false},"excerpt":{"rendered":" Ghi ch\u00fa: T\u00e0i li\u1ec7u tr\u00edch d\u1eabn \u0110\u01b0\u1eddng tr\u00f2n l\u00e0 m\u1ed9t trong nh\u1eefng h\u00ecnh h\u1ecdc kh\u00f3 v\u1edbi nhi\u1ec1u d\u1ea1ng b\u00e0i t\u1eadp kh\u00e1c nhau. \u0110\u1ec3 c\u00f3 th\u1ec3 hi\u1ec3u \u0111\u01b0\u1ee3c \u0111\u1ea7y \u0111\u1ee7 v\u1ec1 h\u00ecnh h\u1ecdc n\u00e0y c\u1ea7n x\u00e1c \u0111\u1ecbnh \u0111\u01b0\u1ee3c l\u00fd thuy\u1ebft c\u0169ng nh\u01b0 b\u00e0i t\u1eadp v\u1ec1 \u0111\u01b0\u1eddng k\u00ednh v\u00e0 d\u00e2y c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n. Ch\u00ednh v\u00ec v\u1eady h\u00e3y […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_disable_autopaging":false},"categories":[9,4],"tags":[],"class_list":["post-997","post","type-post","status-publish","format-standard","hentry","category-toan-hoc","category-trung-hoc"],"_links":{"self":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts\/997","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/comments?post=997"}],"version-history":[{"count":0,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts\/997\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/media?parent=997"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/categories?post=997"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/tags?post=997"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}