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{"id":19744,"date":"2018-03-08T18:39:54","date_gmt":"2018-03-08T11:39:54","guid":{"rendered":"https:\/\/lop12.edu.vn\/?p=19744"},"modified":"2018-03-08T18:48:06","modified_gmt":"2018-03-08T11:48:06","slug":"tap-cuoi-nam-hinh-hoc-12","status":"publish","type":"post","link":"https:\/\/lop12.edu.vn\/tap-cuoi-nam-hinh-hoc-12\/","title":{"rendered":"\u00d4n t\u1eadp cu\u1ed1i n\u0103m H\u00ecnh h\u1ecdc 12"},"content":{"rendered":"

B\u00e0i 1\u00a0<\/span><\/p>\n

Cho l\u0103ng tr\u1ee5 l\u1ee5c gi\u00e1c \u0111\u1ec1u ABCDEF.A’B’C’D’E’F’. O v\u00e0 O’ l\u00e0 t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp hai \u0111\u00e1y, m\u1eb7t ph\u1eb3ng (P) \u0111i qua trung \u0111i\u1ec3m c\u1ee7a OO’ v\u00e0 c\u1eaft c\u00e1c c\u1ea1nh b\u00ean c\u1ee7a l\u0103ng tr\u1ee5. Ch\u1ee9ng minh r\u1eb1ng (P) c\u1ee7a l\u0103ng tr\u1ee5 \u0111\u00e3 cho th\u00e0nh hai \u0111a di\u1ec7n c\u00f3 th\u1ec3 t\u00edch b\u1eb1ng nhau.<\/p>\n

L\u1edcI GI\u1ea2I:<\/strong><\/p>\n

\"\"<\/p>\n

G\u1ecdi I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a OO’<\/p>\n

ABCDEF.A’B’C’D’E’F’ l\u00e0 h\u00ecnh l\u0103ng tr\u1ee5 l\u1ee5c gi\u00e1c \u0111\u1ec1u n\u00ean I l\u00e0 t\u00e2m \u0111\u1ed1i x\u1ee9ng c\u1ee7a c\u00e1c h\u00ecnh ch\u1eef nh\u1eadt ADD’A’, BEE’B’, CFF’C’. V\u1eady n\u1ebfu mp(P) \u0111i qua I v\u00e0 c\u1eaft c\u00e1c c\u1ea1nh AA’, BB’, CC’, DD’, EE’, FF’ theo th\u1ee9 t\u1ef1 t\u1ea1i c\u00e1c \u0111i\u1ec3m M, N, P, Q, R, S th\u00ec I l\u00e0 trung \u0111i\u1ec3m c\u1ee7a MQ, NR v\u00e0 PS<\/p>\n

Suy ra ph\u00e9p \u0111\u1ed1i x\u1ee9ng qua \u0111i\u1ec3m I bi\u1ebfn ABCDEF.MNPQRS th\u00e0nh D’E’F’A’B’C’.QRSMNP.<\/p>\n

Ngh\u0129a l\u00e0 ABCDEF.MNPQRS v\u00e0 D’E’F’A’B’C’. QRSMNP l\u00e0 hai kh\u1ed1i da \u0111i\u1ec7n b\u1eb1ng nhau.<\/p>\n

V\u1eady hai kh\u1ed1i \u0111a di\u1ec7n n\u00f3i tr\u00ean c\u00f3 th\u1ec3 t\u00edch b\u1eb1ng nhau.<\/p>\n

B\u00e0i 2\u00a0<\/span><\/p>\n

Cho kh\u1ed1i l\u1eadp ph\u01b0\u01a1ng ABCD.A’B’C’D’ c\u1ea1nh b\u1eb1ng a. G\u1ecdi E v\u00e0 F l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a B’C’ v\u00e0 C’D’. M\u1eb7t ph\u1eb3ng (AEF) chia kh\u1ed1i l\u1eadp ph\u01b0\u01a1ng \u0111\u00f3 th\u00e0nh hai kh\u1ed1i \u0111a di\u1ec7n (H) v\u00e0 (H’) trong \u0111\u00f3 (H) l\u00e0 kh\u1ed1i \u0111a di\u1ec7n ch\u1ee9a \u0111\u1ec9nh A’. T\u00ednh th\u1ec3 t\u00edch c\u1ee7a (H).<\/p>\n

L\u1edcI GI\u1ea2I<\/strong><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

B\u00e0i 3\u00a0<\/span><\/p>\n

Cho m\u1eb7t c\u1ea7u (S) t\u00e2m O b\u00e1n k\u00ednh r. H\u00ecnh n\u00f3n c\u00f3 \u0111\u01b0\u1eddng tr\u00f2n \u0111\u00e1y (C) v\u00e0 \u0111\u1ec9nh I thu\u1ed9c (S) \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 h\u00ecnh n\u00f3n n\u1ed9i ti\u1ebfp m\u1eb7t c\u1ea7u (S). G\u1ecdi h l\u00e0 chi\u1ec1u cao c\u1ee7a h\u00ecnh n\u00f3n \u0111\u00f3.<\/p>\n

a) Th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i n\u00f3n theo r v\u00e0 h.<\/p>\n

b) X\u00e1c \u0111\u1ecbnh h \u0111\u1ec3 th\u1ec3 t\u00edch c\u1ee7a h\u00ecnh n\u00f3n l\u00e0 l\u1edbn nh\u1ea5t.<\/p>\n

L\u1edcI GI\u1ea2I<\/strong><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

B\u00e0i 4<\/span><\/p>\n

Trong kh\u00f4ng gian Oxyz cho hai \u0111i\u1ec3m A(1; 2; -1), B(7; -2; 3) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng d c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh<\/p>\n

\"\"<\/p>\n

L\u1edcI GI\u1ea2I:<\/strong><\/p>\n

 <\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

B\u00e0i 5<\/span><\/p>\n

Cho t\u1ee9 di\u1ec7n ABCD c\u00f3 c\u1ea1nh AD vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (ABC). Bi\u1ebft r\u1eb1ng AC = AD = 4cm, AB = 3cm, BC = 5cm.<\/p>\n

a)T\u00ednh th\u1ec3 t\u00edch t\u1ee9 di\u1ec7n ABCD<\/p>\n

b)T\u00ednh kho\u1ea3ng c\u00e1ch t\u1eeb \u0111i\u1ec3m A t\u1edbi m\u1eb7t ph\u1eb3ng (BCD)<\/p>\n

L\u1edcI GI\u1ea2I<\/strong><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

B\u00e0i 6<\/span>\u00a0<\/b><\/p>\n

Trong kh\u00f4ng gian Oxyz, cho m\u1eb7t c\u1ea7u (S) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh: x2<\/sup>+ y2<\/sup>\u00a0+ z2<\/sup>\u00a0= 4a2<\/sup>\u00a0(a > 0).<\/p>\n

a)T\u00ednh di\u1ec7n t\u00edch c\u1ee7a m\u1eb7t c\u1ea7u (S) v\u00e0 th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i c\u1ea7u t\u01b0\u01a1ng \u1ee9ng.<\/p>\n

b)M\u1eb7t c\u1ea7u (S) c\u1eaft m\u1eb7t ph\u1eb3ng (Oxy) theo \u0111\u01b0\u1eddng tr\u00f2n (C). X\u00e1c \u0111\u1ecbnh t\u00e2m v\u00e0 b\u00e1n k\u00ednh c\u1ee7a (C).<\/p>\n

c)T\u00ednh di\u1ec7n t\u00edch xung quanh c\u1ee7a h\u00ecnh l\u0103ng tr\u1ee5 nh\u00e2n (C) l\u00e0m \u0111\u00e1y v\u00e0 c\u00f3 chi\u1ec1u cao b\u1eb1ng a . T\u00ednh th\u1ec3 t\u00edch c\u1ee7a kh\u1ed1i tr\u1ee5 t\u01b0\u01a1ng \u1ee9ng.<\/p>\n

L\u1edcI GI\u1ea2I<\/strong><\/p>\n

\"\"<\/p>\n

B\u00e0i 7 <\/span><\/p>\n

Trong kh\u00f4ng gian Oxyz cho hai \u0111\u01b0\u1eddng th\u1eb3ng:<\/p>\n

L\u1edcI GI\u1ea2I<\/strong><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

B\u00e0i 8 <\/strong><\/span><\/p>\n

Trong kh\u00f4ng gian Oxyz cho c\u00e1c \u0111i\u1ec3m A(1; 0; -1), B(3; 4; -2), C(4; -1; 1), D(3; 0; 3)<\/p>\n

a) Ch\u1ee9ng minh r\u1eb1ng A, B, C, D kh\u00f4ng \u0111\u1ed3ng ph\u1eb3ng<\/p>\n

b) Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (ABC) v\u00e0 t\u00ednh kho\u1ea3ng c\u00e1ch t\u1eeb D \u0111\u1ebfn (ABC).<\/p>\n

c) Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t c\u1ea7u ngo\u1ea1i ti\u1ebfp t\u1ee9 di\u1ec7n ABCD.<\/p>\n

d) T\u00ednh th\u1ec3 t\u00edch t\u1ee9 di\u1ec7n ABCD.<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

B\u00e0i 9 <\/b><\/span><\/p>\n

Trong kh\u00f4ng gian Oxyz cho b\u1ed1n \u0111i\u1ec3m A(2; 4; -1), B (1; 4; -1), C(2; 4; 3), D(2; 2; -1)<\/p>\n

a) Ch\u1ee9ng minh r\u1eb1ng c\u00e1c \u0111\u01b0\u1eddng th\u1eb3ng AB, AC, AD vu\u00f4ng g\u00f3c v\u1edbi nhau t\u1eebng \u0111\u00f4i m\u1ed9t. T\u00ednh th\u1ec3 t\u00edch kh\u1ed1i t\u1ee9 di\u1ec7n ABCD.<\/p>\n

b) Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t c\u1ea7u (S) \u0111i qua b\u1ed1n \u0111i\u1ec3m A, B, C, D<\/p>\n

c) Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (\u03b1) ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t c\u1ea7u (S) v\u00e0 song song v\u1edbi mp(ABD).<\/p>\n

L\u1edcI GI\u1ea2I<\/strong><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

B\u00e0i 10 <\/b><\/span><\/p>\n

Trong kh\u00f4ng gian Oxyz cho \u0111\u01b0\u1eddng th\u1eb3ng …<\/p>\n

\"\"<\/p>\n

L\u1edcI GI\u1ea2I<\/strong><\/p>\n

\"\"<\/p>\n

B\u00e0i 11 <\/b><\/span><\/p>\n

Trong kh\u00f4ng gian Oxyz cho c\u00e1c \u0111i\u1ec3m A(-1; 2; 0); B(-3; 0; 2), C(1; 2; 3), D(0; 3; -2)<\/p>\n

a)Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (ABC) v\u00e0 ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng AD.<\/p>\n

b)Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (\u03b1) ch\u1ee9a AD v\u00e0 song song v\u1edbi BC.<\/p>\n

L\u1edcI GI\u1ea2I<\/strong><\/p>\n

\"\"<\/p>\n

B\u00e0i 12 <\/b><\/span><\/p>\n

Trong kh\u00f4ng gian Oxyz cho b\u1ed1n \u0111i\u1ec3m A(3; -2; -2), B(3; 2; 0), C(0; 2; 1) v\u00e0 D(-1; 1; 2)<\/p>\n

a)Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (BCD). Suy ra ABCD l\u00e0 m\u1ed9t t\u1ee9 di\u1ec7n<\/p>\n

b)Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t c\u1ea7u (S) t\u00e2m A v\u00e0 ti\u1ebfp x\u00fac v\u1edbi m\u1eb7t ph\u1eb3ng (BCD)<\/p>\n

c)T\u00ecm t\u1ecda \u0111\u1ed9 ti\u1ebfp \u0111i\u1ec3m c\u1ee7a (S) v\u00e0 m\u1eb7t ph\u1eb3ng (BCD)<\/p>\n

L\u1edcI GI\u1ea2I<\/strong><\/p>\n

\"\"<\/p>\n

B\u00e0i 13 <\/b><\/span><\/p>\n

Trong kh\u00f4ng gian Oxyz cho hai \u0111\u01b0\u1eddng th\u1eb3ng …<\/p>\n

\"\"<\/p>\n

L\u1edcI GI\u1ea2I<\/strong><\/p>\n

\"\"<\/p>\n

B\u00e0i 14<\/b><\/span><\/p>\n

Trong kh\u00f4ng gian cho ba \u0111i\u1ec3m A, B, C.<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

B\u00e0i 15<\/b><\/span><\/p>\n

Cho hai \u0111\u01b0\u1eddng th\u1eb3ng ch\u00e9o nhau:<\/p>\n

\"\"<\/p>\n

a)Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh c\u00e1c m\u1eb7t ph\u1eb3ng (\u03b1) v\u00e0 ( \u03b2) song song v\u1edbi nhau v\u00e0 l\u1ea7n l\u01b0\u1ee3t ch\u1ee9a d v\u00e0 d’.<\/p>\n

b)L\u1ea5y hai \u0111i\u1ec3m M(2; -1; 1) v\u00e0 M'(2; 0; 1) l\u1ea7n l\u01b0\u1ee3t tr\u00ean d v\u00e0 d’. T\u00ednh kho\u1ea3ng c\u00e1ch t\u1eeb M \u0111\u1ebfn m\u1eb7t ph\u1eb3ng ( \u03b2) v\u00e0 kho\u1ea3ng c\u00e1ch t\u1eeb M’ \u0111\u1ebfn m\u1eb7t ph\u1eb3ng (\u03b1). So s\u00e1nh hai kho\u1ea3ng c\u00e1ch \u0111\u00f3.<\/p>\n

L\u1edcI GI\u1ea2I<\/strong><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

B\u00e0i 16 <\/b><\/span><\/p>\n

Trong kh\u00f4ng gian Oxyz cho m\u1eb7t ph\u1eb3ng (\u03b1) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh 4x + y + 2z + 1 =0 v\u00e0 m\u1eb7t ph\u1eb3ng ( \u03b2) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh 2x \u2013 2y + z + 3 = 0<\/p>\n

a) Ch\u1ee9ng minh r\u1eb1ng (\u03b1) c\u1eaft ( \u03b2)<\/p>\n

b) Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh tham s\u1ed1 c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng d l\u00e0 giao c\u1ee7a (\u03b1) v\u00e0 ( \u03b2)<\/p>\n

c) T\u00ecm \u0111i\u1ec3m M’ l\u00e0 \u1ea3nh c\u1ee7a M(4; 2; 1) qua ph\u00e9p \u0111\u1ed1i x\u1ee9ng qua m\u1eb7t ph\u1eb3ng (\u03b1).<\/p>\n

d) T\u00ecm \u0111i\u1ec3m N’ l\u00e0 \u1ea3nh c\u1ee7a N(0; 2; 4) qua ph\u00e9p \u0111\u1ed1i x\u1ee9ng qua \u0111\u01b0\u1eddng th\u1eb3ng d.<\/p>\n

 <\/p>\n

L\u1edcI GI\u1ea2I<\/strong><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

B\u00e0i 1\u00a0 Cho l\u0103ng tr\u1ee5 l\u1ee5c gi\u00e1c \u0111\u1ec1u ABCDEF.A’B’C’D’E’F’. O v\u00e0 O’ l\u00e0 t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp hai \u0111\u00e1y, m\u1eb7t ph\u1eb3ng (P) \u0111i qua trung \u0111i\u1ec3m c\u1ee7a OO’ v\u00e0 c\u1eaft c\u00e1c c\u1ea1nh b\u00ean c\u1ee7a l\u0103ng tr\u1ee5. Ch\u1ee9ng minh r\u1eb1ng (P) c\u1ee7a l\u0103ng tr\u1ee5 \u0111\u00e3 cho th\u00e0nh hai \u0111a di\u1ec7n c\u00f3 th\u1ec3 t\u00edch b\u1eb1ng nhau. […]<\/p>\n","protected":false},"author":3,"featured_media":19745,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_mi_skip_tracking":false,"tdm_status":"","tdm_grid_status":""},"categories":[1298],"tags":[1356,1355,1357],"yoast_head":"\n\u00d4n t\u1eadp cu\u1ed1i n\u0103m H\u00ecnh h\u1ecdc 12<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/lop12.edu.vn\/tap-cuoi-nam-hinh-hoc-12\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u00d4n t\u1eadp cu\u1ed1i n\u0103m H\u00ecnh h\u1ecdc 12\" \/>\n<meta property=\"og:description\" content=\"B\u00e0i 1\u00a0 Cho l\u0103ng tr\u1ee5 l\u1ee5c gi\u00e1c \u0111\u1ec1u ABCDEF.A’B’C’D’E’F’. O v\u00e0 O’ l\u00e0 t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp hai \u0111\u00e1y, m\u1eb7t ph\u1eb3ng (P) \u0111i qua trung \u0111i\u1ec3m c\u1ee7a OO’ v\u00e0 c\u1eaft c\u00e1c c\u1ea1nh b\u00ean c\u1ee7a l\u0103ng tr\u1ee5. Ch\u1ee9ng minh r\u1eb1ng (P) c\u1ee7a l\u0103ng tr\u1ee5 \u0111\u00e3 cho th\u00e0nh hai \u0111a di\u1ec7n c\u00f3 th\u1ec3 t\u00edch b\u1eb1ng nhau. 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