![\"\"](\"https:\/\/lop12.edu.vn\/wp-content\/uploads\/2018\/05\/bai-2-trang-120-sgk-hinh-hoc-11.png\")
<\/p>\nB\u00e0i 1 (trang 120 SGK H\u00ecnh h\u1ecdc 11):<\/b>\u00a0Nh\u1eafc l\u1ea1i \u0111\u1ecbnh ngh\u0129a vect\u01a1 kh\u00f4ng gian.<\/p>\n
Cho h\u00ecnh l\u0103ng tr\u1ee5 tam gi\u00e1c ABC.A’B’C’. H\u00e3y k\u1ec3 t\u00ean nh\u1eefng vect\u01a1 b\u1eb1ng vect\u01a1 c\u00f3 \u0111i\u1ec3m \u0111\u1ea7u v\u00e0 \u0111i\u1ec3m cu\u1ed1i l\u00e0 \u0111\u1ec9nh c\u1ee7a h\u00ecnh l\u0103ng tr\u1ee5.<\/p>\n
L\u1eddi gi\u1ea3i:<\/b><\/p>\n
Vect\u01a1 trong kh\u00f4ng gian l\u00e0 m\u1ed9t \u0111o\u1ea1n th\u1eb3ng c\u00f3 \u0111\u1ecbnh h\u01b0\u1edbng, t\u1ee9c l\u00e0 m\u1ed9t \u0111o\u1ea1n th\u1eb3ng \u0111\u00e3 \u0111\u01b0\u1ee3c ch\u1ec9 r\u00f5 \u0111i\u1ec3m \u0111\u1ea7u v\u00e0 \u0111i\u1ec3m cu\u1ed1i.<\/p>\n
V\u00ec c\u00e1c c\u1ea1nh b\u00ean c\u1ee7a h\u00ecnh l\u0103ng tr\u1ee5 l\u00e0 c\u00e1c \u0111o\u1ea1n th\u1eb3ng song song v\u00e0 b\u1eb1ng nhau n\u00ean c\u00e1c vect\u01a1 b\u1eb1ng vect\u01a1 v\u00e0 c\u00f3 \u0111i\u1ec3m \u0111\u1ea7u v\u00e0 \u0111i\u1ec3m cu\u1ed1i l\u00e0 \u0111\u1ec9nh c\u1ee7a h\u00ecnh l\u0103ng tr\u1ee5 l\u00e0: c\u00e1c vector BB’, CC’, DD’.<\/p>\n
B\u00e0i 2 (trang 120 SGK H\u00ecnh h\u1ecdc 11):<\/b>\u00a0Trong kh\u00f4ng gian cho ba vect\u01a1 a , b v\u00e0 c \u0111\u1ec1u kh\u00e1c vect\u01a1 0 . Khi n\u00e0o ba v\u00e9c t\u01a1 \u0111\u00f3 \u0111\u1ed3ng ph\u1eb3ng?<\/p>\n
L\u1eddi gi\u1ea3i:<\/b><\/p>\n
<\/p>\n
Th\u1ecfa m\u00e3n :<\/p>\n
– Gi\u00e1 c\u1ee7a 3 vector \u0111\u1ec1u song song v\u1edbi m\u1eb7t ph\u1eb3ng (P) n\u00ean ch\u00fang \u0111\u1ed3ng ph\u1eb3ng<\/p>\n
– Khi ba vect\u01a1 c\u00f3 gi\u00e1 c\u1ee7a ch\u00fang c\u00f9ng song song v\u1edbi m\u1ed9t m\u1eb7t ph\u1eb3ng<\/p>\n
B\u00e0i 3 (trang 120 SGK H\u00ecnh h\u1ecdc 11):<\/b>\u00a0Trong kh\u00f4ng gian hai \u0111\u01b0\u1eddng th\u1eb3ng kh\u00f4ng c\u1eaft nhau c\u00f3 th\u1ec3 vu\u00f4ng g\u00f3c v\u1edbi nhau kh\u00f4ng? Gi\u1ea3 s\u1eed hai \u0111\u01b0\u1eddng th\u1eb3ng a v\u00e0 b l\u1ea7n l\u01b0\u1ee3t c\u00f3 vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng l\u00e0 vector u v\u00e0 vector v . Khi n\u00e0o ta c\u00f3 k\u1ebft lu\u1eadn a v\u00e0 b vu\u00f4ng g\u00f3c v\u1edbi nhau?<\/p>\n
L\u1eddi gi\u1ea3i:<\/b><\/p>\n
Trong kh\u00f4ng gian, hai \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi nhau kh\u00f4ng nh\u1ea5t thi\u1ebft ph\u1ea3i c\u1eaft nhau v\u00ec v\u1eady hai \u0111\u01b0\u1eddng th\u1eb3ng kh\u00f4ng c\u1eaft nhau v\u1eabn c\u00f3 th\u1ec3 vu\u00f4ng g\u00f3c v\u1edbi nhau.<\/p>\n
\u0110\u01b0\u1eddng th\u1eb3ng a c\u00f3 vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng\u00a0u\u2192<\/p>\n
\u0110\u01b0\u1eddng th\u1eb3ng b c\u00f3 vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng\u00a0v\u2192<\/p>\n
a vu\u00f4ng g\u00f3c v\u1edbi b khi v\u00e0 ch\u1ec9 khi t\u00edch v\u00f4 h\u01b0\u1edbng c\u1ee7a hai vect\u01a1 v\u00e0 b\u1eb1ng kh\u00f4ng. a \u22a5 b \u21d4u\u2192\u00a0.v\u2192\u00a0= 0.<\/p>\n
B\u00e0i 4 (trang 120 SGK H\u00ecnh h\u1ecdc 11):<\/b>\u00a0Mu\u1ed1n ch\u1ee9ng minh \u0111\u01b0\u1eddng th\u1eb3ng a vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (\u03b1) c\u00f3 c\u1ea7n ch\u1ee9ng minh a vu\u00f4ng g\u00f3c v\u1edbi m\u1ecdi \u0111\u01b0\u1eddng th\u1eb3ng c\u1ee7a (\u03b1) hay kh\u00f4ng?<\/p>\n
L\u1eddi gi\u1ea3i:<\/b><\/p>\n
Kh\u00f4ng c\u1ea7n ch\u1ee9ng minh a vu\u00f4ng g\u00f3c v\u1edbi m\u1ecdi \u0111\u01b0\u1eddng th\u1eb3ng c\u1ee7a m\u1eb7t ph\u1eb3ng.<\/p>\n
Ta c\u00f3 th\u1ec3 ch\u1ecdn m\u1ed9t trong s\u1ed1 nh\u1eefng c\u00e1ch sau \u0111\u1ec3 ch\u1ee9ng minh \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng<\/p>\n
– C\u00e1ch 1 : Ch\u1ee9ng minh \u0111\u01b0\u1eddng th\u1eb3ng d vu\u00f4ng g\u00f3c v\u1edbi hai \u0111\u01b0\u1eddng th\u1eb3ng c\u1eaft nhau n\u1eb1m trong m\u1eb7t ph\u1eb3ng<\/p>\n
– C\u00e1ch 2 : S\u1eed d\u1ee5ng \u0111\u1ecbnh l\u00ed : “N\u1ebfu hai m\u1eb7t ph\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi nhau th\u00ec b\u1ea5t k\u00ec \u0111\u01b0\u1eddng th\u1eb3ng n\u00e0o n\u1eb1m trong m\u1eb7t ph\u1eb3ng n\u00e0y v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi giao tuy\u1ebfn th\u00ec c\u0169ng vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng kia”.<\/p>\n
– C\u00e1ch 3 : S\u1eed d\u1ee5ng \u0111\u1ecbnh l\u00ed : ” N\u1ebfu hai m\u1eb7t ph\u1eb3ng ph\u00e2n bi\u1ec7t c\u00f9ng vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng th\u1ee9 3 th\u00ec giao tuy\u1ebfn c\u1ee7a ch\u00fang c\u0169ng s\u1ebd vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng \u0111\u00f3”<\/p>\n
B\u00e0i 5 (trang 120 SGK H\u00ecnh h\u1ecdc 11):<\/b>\u00a0Nh\u1eafc l\u1ea1i n\u1ed9i dung \u0111\u1ecbnh l\u00ed ba \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c<\/p>\n
L\u1eddi gi\u1ea3i:<\/b><\/p>\n
Cho \u0111\u01b0\u1eddng th\u1eb3ng a kh\u00f4ng vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (P) v\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng b n\u1eb1m trong (P) . Khi \u0111\u00f3, \u0111i\u1ec1u ki\u1ec7n c\u1ea7n v\u00e0 \u0111\u1ee7 \u0111\u1ec3 b vu\u00f4ng g\u00f3c v\u1edbi a l\u00e0 b vu\u00f4ng g\u00f3c v\u1edbi h\u00ecnh chi\u1ebfu a’ c\u1ee7a a tr\u00ean (P).<\/p>\n
![\"\"](\"https:\/\/lop12.edu.vn\/wp-content\/uploads\/2018\/05\/bai-5-trang-120-sgk-hinh-hoc-11.png\")
<\/p>\n
B\u00e0i 6 (trang 120 SGK H\u00ecnh h\u1ecdc 11):<\/b>\u00a0Nh\u1eafc l\u1ea1i \u0111\u1ecbnh ngh\u0129a:<\/p>\n
a) G\u00f3c gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng v\u00e0 m\u1eb7t ph\u1eb3ng.<\/p>\n
b) G\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng.<\/p>\n
L\u1eddi gi\u1ea3i:<\/b><\/p>\n
a) \u0110\u1ecbnh ngh\u0129a: G\u00f3c gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng v\u00e0 m\u1eb7t ph\u1eb3ng.<\/p>\n
Cho \u0111\u01b0\u1eddng th\u1eb3ng d c\u1eaft m\u1eb7t ph\u1eb3ng (\u03b1) t\u1ea1i \u0111i\u1ec3m O v\u00e0 d kh\u00f4ng vu\u00f4ng g\u00f3c v\u1edbi (\u03b1). G\u00f3c gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng d v\u00e0 m\u1eb7t ph\u1eb3ng (\u03b1) l\u00e0 g\u00f3c t\u1ea1o b\u1edfi \u0111\u01b0\u1eddng th\u1eb3ng d v\u00e0 h\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c g\u00f3c d‘<\/sup>\u00a0c\u1ee7a d tr\u00ean m\u1eb7t ph\u1eb3ng (\u03b1), k\u00ed hi\u1ec7u g\u00f3c (d,\u03b1).<\/p>\n – N\u1ebfu d vu\u00f4ng g\u00f3c g\u00f3c v\u1edbi (\u03b1) ta qui \u01b0\u1edbc g\u00f3c (d,\u03b1) = 90o<\/sup>.<\/p>\n – N\u1ebfu d \/\/ (\u03b1) hay d n\u1eb1m trong (\u03b1) ta quy \u01b0\u1edbc g\u00f3c (d,\u03b1) = 90o<\/sup>.<\/p>\n b) \u0110\u1ecbnh ngh\u0129a: G\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng<\/p>\n Gi\u1ea3 s\u1eed hai m\u1eb7t ph\u1eb3ng (\u03b1) v\u00e0 (\u03b2) c\u1eaft nhau theo giao tuy\u1ebfn c. T\u1eeb \u0111i\u1ec3m I b\u1ea5t k\u00ec tr\u00ean c ta d\u1ef1ng trong (\u03b1) \u0111\u01b0\u1eddng th\u1eb3ng a vu\u00f4ng g\u00f3c v\u1edbi c v\u00e0 d\u1ef1ng trong (\u03b2) \u0111\u01b0\u1eddng th\u1eb3ng b vu\u00f4ng g\u00f3c v\u1edbi c. Ta g\u1ecdi g\u00f3c gi\u1eefa hai \u0111\u01b0\u1eddng a v\u00e0 b l\u00e0 g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (\u03b1) v\u00e0 (\u03b2). Nh\u01b0 v\u1eady g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (\u03b1) v\u00e0 (\u03b2) lu\u00f4n c\u00f3 s\u1ed1 \u0111o b\u00e9 h\u01a1n ho\u1eb7c b\u1eb1ng 90o<\/sup>.<\/p>\n *N\u1ebfu hai m\u1eb7t ph\u1eb3ng song song ho\u1eb7c tr\u00f9ng v\u1edbi nhau th\u00ec ta n\u00f3i r\u1eb1ng g\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng \u0111\u00f3 b\u1eb1ng 0o<\/sup>. G\u00f3c gi\u1eefa hai m\u1eb7t ph\u1eb3ng (\u03b1) v\u00e0 (\u03b2) \u0111\u01b0\u1ee3c k\u00ed hi\u1ec7u l\u00e0 (\u03b1, \u03b2), ta c\u00f3 0o<\/sup>\u00a0\u2264 (\u03b1, \u03b2) \u2264 90o<\/sup>.<\/p>\n B\u00e0i 7 (trang 120 SGK C\u00e2u h\u1ecfi H\u00ecnh h\u1ecdc 11):<\/b>\u00a0Mu\u1ed1n ch\u1ee9ng minh m\u1eb7t ph\u1eb3ng (\u03b1) vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (\u03b2) ta c\u00f3 th\u1ec3 ?<\/p>\n L\u1eddi gi\u1ea3i:<\/b><\/p>\n Ch\u1ee9ng minh (\u03b1) ch\u1ee9a m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi (\u03b2) ho\u1eb7c (\u03b2) ch\u1ee9a m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi (\u03b1).<\/p>\n Ho\u1eb7c ch\u1ee9ng minh g\u00f3c gi\u1eefa (\u03b1) v\u00e0 (\u03b2) b\u1eb1ng 90o<\/sup>.<\/p>\n B\u00e0i 8 (trang 120 SGK H\u00ecnh h\u1ecdc 11):<\/b>\u00a0H\u00e3y n\u00eau c\u00e1ch t\u00ednh kho\u1ea3ng c\u00e1ch :<\/p>\n a) T\u1eeb m\u1ed9t \u0111i\u1ec3m \u0111\u1ebfn m\u1ed9t \u0111\u01b0\u1eddng th\u1eb3ng ;<\/p>\n b) T\u1eeb \u0111\u01b0\u1eddng th\u1eb3ng a \u0111\u1ebfn m\u1eb7t ph\u1eb3ng (\u03b1) song song v\u1edbi a ;<\/p>\n c) Gi\u1eefa hai m\u1eb7t ph\u1eb3ng song song.<\/p>\n L\u1eddi gi\u1ea3i:<\/b><\/p>\n a) \u0110\u1ec3 t\u00ednh kho\u1ea3ng c\u00e1ch t\u1eeb \u0111i\u1ec3m O \u0111\u1ebfn \u0111\u01b0\u1eddng th\u1eb3ng \u0394 kh\u00f4ng \u0111i qua O, ta x\u00e1c \u0111\u1ecbnh m\u1eb7t ph\u1eb3ng (O; \u0394) v\u00e0 trong m\u1eb7t ph\u1eb3ng n\u00e0y k\u1ebb OH \u22a5 \u0394. \u0110\u1ed9 d\u00e0i OH ch\u00ednh l\u00e0 kho\u1ea3ng c\u00e1ch t\u1eeb O \u0111\u1ebfn \u0394.<\/p>\n b) \u0110\u1ec3 t\u00ednh kho\u1ea3ng c\u00e1ch gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng a v\u00e0 mp(P) song song v\u1edbi (P), ta l\u1ea5y m\u1ed9t \u0111i\u1ec3m M b\u1ea5t k\u00ec thu\u1ed9c \u0111\u01b0\u1eddng th\u1eb3ng a. Kho\u1ea3ng c\u00e1ch MN t\u1eeb \u0111i\u1ec3m M \u0111\u1ebfn mp(P) ch\u00ednh l\u00e0 kho\u1ea3ng c\u00e1ch gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng v\u00e0 mp(P) song song v\u1edbi a.<\/p>\n c) \u0110\u1ec3 t\u00ednh kho\u1ea3ng c\u00e1ch gi\u1eefa hai mp(P) v\u00e0 (P‘<\/sup>) song song v\u1edbi nhau, ta l\u1ea5y m\u1ed9t \u0111i\u1ec3m M thu\u1ed9c (P) v\u00e0 t\u00ecm kho\u1ea3ng c\u00e1ch MH t\u1eeb \u0111i\u1ec3m M \u0111\u1ebfn mp(P‘<\/sup>).<\/p>\n B\u00e0i 9 (trang 120 SGK H\u00ecnh h\u1ecdc 11):<\/b>\u00a0Cho a v\u00e0 b l\u00e0 hai \u0111\u01b0\u1eddng th\u1eb3ng ch\u00e9o nhau. C\u00f3 th\u1ec3 t\u00ednh kho\u1ea3ng c\u00e1ch gi\u1eefa hai \u0111\u01b0\u1eddng th\u1eb3ng ch\u00e9o nhau n\u00e0y b\u1eb1ng nh\u1eefng c\u00e1ch n\u00e0o?<\/p>\n L\u1eddi gi\u1ea3i:<\/b><\/p>\n C\u00f3 2 c\u00e1ch \u0111\u1ec3 t\u00ednh kho\u1ea3ng c\u00e1ch gi\u1eefa hai \u0111\u01b0\u1eddng th\u1eb3ng ch\u00e9o nhau<\/b><\/p>\n B\u00e0i 10 (trang 120 SGK H\u00ecnh h\u1ecdc 11):<\/b>\u00a0Ch\u1ee9ng minh r\u1eb1ng t\u1eadp h\u1ee3p c\u00e1c \u0111i\u1ec3m c\u00e1ch \u0111\u1ec1u ba \u0111\u1ec9nh c\u1ee7a m\u1ed9t tam gi\u00e1c ABC l\u00e0 \u0111\u01b0\u1eddng vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng (ABC) v\u00e0 \u0111i qua t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c ABC.<\/p>\n L\u1eddi gi\u1ea3i:<\/b><\/p>\n L\u1ea5y m\u1ed9t \u0111i\u1ec3m M b\u1ea5t k\u00ec trong kh\u00f4ng gian sao cho MA = MB = MC. T\u1eeb M k\u1ebb MO vu\u00f4ng g\u00f3c v\u1edbi mp(ABC). C\u00e1c tam gi\u00e1c vu\u00f4ng MOA, MOB, MOC b\u1eb1ng nhau, cho ta OA = OB = OC.<\/p>\n – Suy ra O l\u00e0 t\u00e2m \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c ABC. V\u1eady c\u00e1c \u0111i\u1ec3m M c\u00e1ch \u0111\u1ec1u ba \u0111\u1ec9nh c\u1ee7a tam gi\u00e1c ABC n\u1eb1m tr\u00ean \u0111\u01b0\u1eddng th\u1eb3ng d \u0111i qua t\u00e2m O c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i ti\u1ebfp tam gi\u00e1c ABC v\u00e0 vu\u00f4ng g\u00f3c v\u1edbi mp(ABC). Ng\u01b0\u1ee3c l\u1ea1i, l\u1ea5y m\u1ed9t \u0111i\u1ec3m M‘<\/sup>\u00a0\u2208 d, n\u1ed1i M‘<\/sup>A, M‘<\/sup>B, M‘<\/sup>C.<\/p>\n – Do M‘<\/sup>O chung v\u00e0 OA = OB = OC n\u00ean c\u00e1c tam gi\u00e1c vu\u00f4ng M‘<\/sup>OA, M‘<\/sup>OB, M‘<\/sup>OC b\u1eb1ng nhau, cho ta M‘<\/sup>A = M‘<\/sup>B = M‘<\/sup>C.<\/p>\n – T\u1ee9c l\u00e0 \u0111i\u1ec3m M‘<\/sup>\u00a0c\u00e1ch \u0111\u1ec1u ba \u0111\u1ec9nh A, B, C c\u1ee7a tam gi\u00e1c ABC.<\/p>\n – K\u1ebft lu\u1eadn : T\u1eadp h\u1ee3p c\u00e1c \u0111i\u1ec3m c\u00e1ch \u0111\u1ec1u ba \u0111\u1ec9nh c\u1ee7a tam gi\u00e1c ABC l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi mp(ABC) v\u00e0 \u0111i qua t\u00e2m c\u1ee7a \u0111\u01b0\u1eddng tr\u00f2n ngo\u1ea1i tam gi\u00e1c ABC.<\/p>\n<\/div>\n<\/div>\n <\/p>\n","protected":false},"excerpt":{"rendered":" B\u00e0i 1 (trang 120 SGK H\u00ecnh h\u1ecdc 11):\u00a0Nh\u1eafc l\u1ea1i \u0111\u1ecbnh ngh\u0129a vect\u01a1 kh\u00f4ng gian. Cho h\u00ecnh l\u0103ng tr\u1ee5 tam gi\u00e1c ABC.A’B’C’. H\u00e3y k\u1ec3 t\u00ean nh\u1eefng vect\u01a1 b\u1eb1ng vect\u01a1 c\u00f3 \u0111i\u1ec3m \u0111\u1ea7u v\u00e0 \u0111i\u1ec3m cu\u1ed1i l\u00e0 \u0111\u1ec9nh c\u1ee7a h\u00ecnh l\u0103ng tr\u1ee5. L\u1eddi gi\u1ea3i: Vect\u01a1 trong kh\u00f4ng gian l\u00e0 m\u1ed9t \u0111o\u1ea1n th\u1eb3ng c\u00f3 \u0111\u1ecbnh h\u01b0\u1edbng, t\u1ee9c l\u00e0 […]<\/p>\n","protected":false},"author":3,"featured_media":29393,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_mi_skip_tracking":false,"tdm_status":"","tdm_grid_status":""},"categories":[1318],"tags":[1464,1463],"yoast_head":"\n![\"\"](\"https:\/\/lop12.edu.vn\/wp-content\/uploads\/2018\/05\/bai-7-trang-120-sgk-cau-hoi-hinh-hoc-11.png\")
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