Tính chất ba đường cao của tam giác- Level3

Home Lớp 7 Toán lớp 7 - Sách kết nối tri thức Tính chất ba đường cao của tam giácBài tập nâng cao

{"common":{"save":0,"post_id":"1509","level":3,"total":10,"point":10,"point_extra":0},"segment":[{"id":"2001","post_id":"1509","mon_id":"0","chapter_id":"0","question":"L\u1ef1a ch\u1ecdn \u0111\u00e1p \u00e1n \u0111\u00fang nh\u1ea5t","options":{"time":24,"part":[{"title":"L\u1ef1a ch\u1ecdn \u0111\u00e1p \u00e1n \u0111\u00fang nh\u1ea5t","title_trans":"","temp":"multiple_choice","correct":[[4]],"list":[{"point":10,"ques":" $\\triangle{ABC}$ vu\u00f4ng t\u1ea1i $A$, \u0111\u01b0\u1eddng cao $AH$. L\u1ea5y \u0111i\u1ec3m $I$ l\u00e0 trung \u0111i\u1ec3m c\u1ee7a $AC$, $K$ l\u00e0 trung \u0111i\u1ec3m c\u1ee7a $AH$, $D$ l\u00e0 trung \u0111i\u1ec3m c\u1ee7a $HC$. Kh\u1eb3ng \u0111\u1ecbnh n\u00e0o sau \u0111\u00e2y \u0111\u00fang? ","select":["A. $I$ l\u00e0 giao \u0111i\u1ec3m ba \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a $\\triangle{AHC}$ ","B. $KD \/\/ AC$","C. $BK \\perp AD$ ","D. C\u1ea3 ba \u0111\u00e1p \u00e1n tr\u00ean \u0111\u1ec1u \u0111\u00fang"],"hint":"","explain":" <center><img src='img\/H7C3B23_K1.png' \/><\/center><br\/> $\\blacktriangleright$ $\\triangle{AHC}$ vu\u00f4ng t\u1ea1i $H$ c\u00f3 $I$ l\u00e0 trung \u0111i\u1ec3m c\u1ee7a $AC$ <br\/> $\\Rightarrow$ $HI$ l\u00e0 \u0111\u01b0\u1eddng trung tuy\u1ebfn $\\Rightarrow$ $HI = AI = IC$ (t\u00ednh ch\u1ea5t trung tuy\u1ebfn trong tam gi\u00e1c vu\u00f4ng) <br\/> $IA = IH$ $\\Rightarrow$ $I$ n\u1eb1m tr\u00ean \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a $AH$ (1) <br\/> $IH = IC$ $\\Rightarrow$ $I$ n\u1eb1m tr\u00ean \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a $HC$ (2) <br\/> $IA = IC$ $\\Rightarrow$ $I$ n\u1eb1m tr\u00ean \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a $AC$ (3) <br\/> T\u1eeb (1), (2), (3) $\\Rightarrow$ $I$ l\u00e0 giao \u0111i\u1ec3m c\u1ee7a ba \u0111\u01b0\u1eddng trung tr\u1ef1c c\u1ee7a $\\triangle{AHC}$ <br\/> $\\Rightarrow$ \u0110\u00e1p \u00e1n A \u0111\u00fang <\/b> <br\/> $\\blacktriangleright$ X\u00e9t $\\triangle{KHD}$ v\u00e0 $\\triangle{DIK}$ c\u00f3: <br\/> $KD$ chung <br\/> M\u00e0 $AH \\perp BC; DI \\perp BC \\Rightarrow AH \/\/ DI$ $\\Rightarrow$ $\\widehat{HDK} = \\widehat{KDI}$ (so le trong) <br\/> $IK \\perp AH; BC \\perp AH$ $\\Rightarrow$ $KI \/\/ BC$ $\\Rightarrow$ $\\widehat{HKD} = \\widehat{IKD}$ (so le trong) <br\/> V\u1eady $\\triangle{KHD} = \\triangle{DIK}$ (g.c.g) <br\/> $\\Rightarrow$ $HK = ID; HD = KI$ (c\u1eb7p c\u1ea1nh t\u01b0\u01a1ng \u1ee9ng) <br\/> X\u00e9t hai tam gi\u00e1c vu\u00f4ng $KHD$ v\u00e0 $DIC$ c\u00f3: <br\/> $HK = ID$ (cmt) <br\/> $HD = HC$ ($ID$ l\u00e0 trung tr\u1ef1c) <br\/> $\\Rightarrow$ $\\triangle{KHD} = \\triangle{DIC}$ (hai c\u1ea1nh g\u00f3c vu\u00f4ng) <br\/> $\\Rightarrow$ $\\widehat{HDK} = \\widehat{DCI}$ (hai g\u00f3c t\u01b0\u01a1ng \u1ee9ng) <br\/> $\\Rightarrow$ $KD \/\/ AC$ (v\u00ec hai g\u00f3c \u0111\u1ed3ng v\u1ecb b\u1eb1ng nhau) <br\/> $\\Rightarrow$ \u0110\u00e1p \u00e1n B \u0111\u00fang <\/b> <br\/> $\\blacktriangleright$ Ta c\u00f3 $KD \/\/ AC$ (cmt) v\u00e0 $AB \\perp AC$ <br\/> $\\Rightarrow$ $KD \\perp AB$ <br\/> X\u00e9t $\\triangle{ABD}$ c\u00f3: $AH \\perp BD$ (gi\u1ea3 thi\u1ebft) v\u00e0 $KD \\perp AB$ <br\/> N\u00ean $K$ l\u00e0 tr\u1ef1c t\u00e2m c\u1ee7a $\\triangle{ABD}$ <br\/> V\u1eady $KB$ l\u00e0 \u0111\u01b0\u1eddng cao th\u1ee9 ba c\u1ee7a $\\triangle{ABD}$ suy ra $BK \\perp AD$ <br\/> $\\Rightarrow$ \u0110\u00e1p \u00e1n C \u0111\u00fang <\/b> <br\/> V\u1eady \u0111\u00e1p \u00e1n \u0111\u00fang nh\u1ea5t l\u00e0: D <\/span> ","column":2}]}]},"correct":"","level":"3","hint":"","answer":"","type":"json","extra_type":"","time":"0","user_id":"0","test":"0","date":"2019-09-30 09:23:07"},{"id":"2002","post_id":"1509","mon_id":"0","chapter_id":"0","question":"Ch\u1ecdn nh\u1eefng<\/u> \u0111\u00e1p \u00e1n \u0111\u00fang","options":{"time":24,"part":[{"title":"Ch\u1ecdn nh\u1eefng<\/u> \u0111\u00e1p \u00e1n \u0111\u00fang","title_trans":"Ch\u1ecdn \u0111\u01b0\u1ee3c nhi\u1ec1u \u0111\u00e1p \u00e1n","temp":"checkbox","correct":[["1","2","3"]],"list":[{"point":10,"img":"","ques":"Cho tam gi\u00e1c $ABC$, \u0111\u01b0\u1eddng cao $AH$. Tr\u00ean tia \u0111\u1ed1i c\u1ee7a $AH$ l\u1ea5y \u0111i\u1ec3m $D$ sao cho $AD = BC$. T\u1ea1i $B$ k\u1ebb \u0111\u01b0\u1eddng th\u1eb3ng $BE \\perp AB$ ($E$ v\u00e0 $C$ thu\u1ed9c hai n\u1eeda m\u1eb7t ph\u1eb3ng \u0111\u1ed1i nhau b\u1edd l\u00e0 $AB$). T\u1ea1i $C$ k\u1ebb \u0111\u01b0\u1eddng th\u1eb3ng $CF \\perp AC$ v\u00e0 $CF = AC$ ($F$ v\u00e0 $B$ thu\u1ed9c hai n\u1eeda m\u1eb7t ph\u1eb3ng \u0111\u1ed1i nhau b\u1edd $AC$).","hint":"","column":2,"number_true":2,"select":["A. $DC = BF$","B. $DC \\perp BF$","C. Ba \u0111\u01b0\u1eddng th\u1eb3ng $DH, BF, CE$ \u0111\u1ed3ng quy","D. $DC > BF$ "],"explain":" <center><img src='img\/H7C3B23_K2.png' \/><\/center> <br\/> G\u1ecdi $I$ l\u00e0 giao \u0111i\u1ec3m c\u1ee7a $CD$ v\u00e0 $BF$ v\u00e0 $G$ l\u00e0 giao \u0111i\u1ec3m c\u1ee7a $BD$ v\u00e0 $EC$ <br\/> $\\blacktriangleright$ $\\triangle{DAC}$ v\u00e0 $\\triangle{BCF}$ c\u00f3: <br\/> $DA = DC$ (gt) <br\/> $\\widehat{DAC} = \\widehat{BCF}$ (g\u00f3c c\u00f3 c\u1ea1nh t\u01b0\u01a1ng \u1ee9ng vu\u00f4ng g\u00f3c, c\u00f9ng t\u00f9 $CB \\perp AD, CF \\perp AC$) <br\/> $AC = CF$ (gt) <br\/> $\\Rightarrow$ $\\triangle{DAC} = \\triangle{BCF}$ (c.g.c) <br\/> $\\Rightarrow$ $DC = BF$ (hai c\u1ea1nh t\u01b0\u01a1ng \u1ee9ng) \u0110\u00e1p \u00e1n A \u0111\u00fang, D sai <\/b> <br\/> $\\blacktriangleright$ $\\widehat{C_{1}} = \\widehat{F}$ (hai g\u00f3c t\u01b0\u01a1ng \u1ee9ng) <br\/> M\u00e0 $\\widehat{C_{1}} + \\widehat{C_{2}} = 90^{o}$ (gi\u1ea3 thi\u1ebft) <br\/> $\\Rightarrow$ $\\widehat{F} + \\widehat{C_{2}} = 90^{o}$ <br\/> Trong $\\triangle{CFI}$ c\u00f3: $\\widehat{F} + \\widehat{C_{2}} = 90^{o}$ $\\Rightarrow$ $\\widehat{CIF} = 90^{o}$ <br\/> V\u1eady $DC \\perp BF$ (\u0110\u00e1p \u00e1n B \u0111\u00fang) <\/b> <br\/> $\\blacktriangleright$ Ch\u1ee9ng minh t\u01b0\u01a1ng t\u1ef1 ta \u0111\u01b0\u1ee3c $\\triangle{DAB} = \\triangle{CBE}$ (c.g.c) <br\/> $\\Rightarrow$ $\\widehat{B_{1}} = \\widehat{E}$ (hai g\u00f3c t\u01b0\u01a1ng \u1ee9ng) <br\/> M\u00e0 $\\widehat{B_{1}} + \\widehat{B_{2}} = 90^{o}$ $\\Rightarrow$ $\\widehat{E} + \\widehat{B_{2}} = 90^{o}$ <br\/> Trong $\\triangle{EBG}$ c\u00f3: $\\widehat{E} + \\widehat{B_{2}} = 90^{o}$ $\\Rightarrow$ $\\widehat{EBG} = 90^{o}$ <br\/> $\\Rightarrow$ $DB \\perp CE$ <br\/> Trong $\\triangle{DBC}$ c\u00f3 $DH \\perp BC; BI \\perp AC; CG \\perp AB$ <br\/> V\u1eady $DH; BI; CG$ l\u00e0 ba \u0111\u01b0\u1eddng cao c\u1ee7a $\\triangle{DBC}$ hay ba \u0111\u01b0\u1eddng cao \u0111\u1ed3ng quy (t\u00ednh ch\u1ea5t ba \u0111\u01b0\u1eddng cao c\u1ee7a tam gi\u00e1c) <br\/> \u0110\u00e1p \u00e1n C \u0111\u00fang <\/b> <br\/> V\u1eady c\u00e1c \u0111\u00e1p \u00e1n \u0111\u00fang l\u00e0: A, B, C<\/span> "}]}]},"correct":"","level":"3","hint":"","answer":"","type":"json","extra_type":"","time":"0","user_id":"0","test":"0","date":"2019-09-30 09:23:07"},{"id":"2003","post_id":"1509","mon_id":"0","chapter_id":"0","question":"L\u1ef1a ch\u1ecdn \u0111\u00e1p \u00e1n \u0111\u00fang nh\u1ea5t","options":{"time":24,"part":[{"title":"L\u1ef1a ch\u1ecdn \u0111\u00e1p \u00e1n \u0111\u00fang nh\u1ea5t","title_trans":"","temp":"multiple_choice","correct":[[1]],"list":[{"point":10,"ques":" Cho \u0111o\u1ea1n th\u1eb3ng $AB$. Tr\u00ean c\u00f9ng m\u1ed9t n\u1eeda m\u1eb7t ph\u1eb3ng b\u1edd $AB$, v\u1ebd hai tia $Ax, By$ c\u00f9ng vu\u00f4ng g\u00f3c v\u1edbi $AB$. Tr\u00ean hai tia $Ax, By$ l\u1ea7n l\u01b0\u1ee3t l\u1ea5y hai \u0111i\u1ec3m $C, D$ sao cho $AC = \\dfrac{1}{2}BD$. V\u1ebd $BE$ vu\u00f4ng g\u00f3c v\u1edbi $AD$ $(E \\in AD)$ v\u00e0 g\u1ecdi $F$ l\u00e0 trung \u0111i\u1ec3m c\u1ee7a $ED$. Khi \u0111\u00f3 $CF \\perp BF$. \u0110\u00fang <\/b> hay sai <\/b>? ","select":["A. \u0110\u00daNG ","B. SAI"],"hint":"","explain":" H\u01b0\u1edbng d\u1eabn:<\/span><br\/> B\u01b0\u1edbc 1: <\/b> G\u1ecdi $H$ l\u00e0 trung \u0111i\u1ec3m c\u1ee7a $EB$, ch\u1ee9ng minh $AH \\perp BF$ d\u1ef1a v\u00e0o t\u00ednh ch\u1ea5t ba \u0111\u01b0\u1eddng cao trong tam gi\u00e1c $ABF$ <br\/> B\u01b0\u1edbc 2: <\/b> Ch\u1ee9ng minh $CF \/\/ AH$ b\u1eb1ng c\u00e1ch ch\u1ee9ng minh hai g\u00f3c \u1edf v\u1ecb tr\u00ed so le trong b\u1eb1ng nhau <br\/> B\u01b0\u1edbc 3: <\/b> T\u1eeb b\u01b0\u1edbc 1 v\u00e0 b\u01b0\u1edbc 2 suy ra \u0111i\u1ec1u c\u1ea7n ch\u1ee9ng minh <br\/><br\/> B\u00e0i gi\u1ea3i:<\/span><br\/> <center><img src='img\/H7C3B23_K3.png' \/><\/center><br\/> $\\blacktriangleright$ G\u1ecdi $H$ l\u00e0 trung \u0111i\u1ec3m c\u1ee7a $BE$. Khi \u0111\u00f3 $HF$ l\u00e0 \u0111\u01b0\u1eddng trung b\u00ecnh c\u1ee7a $\\triangle{BDE}$ \u1ee9ng v\u1edbi \u0111\u00e1y $BD$ (xem b\u00e0i $64$ s\u00e1ch b\u00e0i t\u1eadp To\u00e1n $7$ t\u1eadp $1$ trang $146$) <br\/> $\\Rightarrow$ $HF \/\/ BD$ v\u00e0 $HF = \\dfrac{1}{2}BD$ (t\u00ednh ch\u1ea5t \u0111\u01b0\u1eddng trung b\u00ecnh) <br\/> M\u00e0 $BD \\perp AB$ (gt) n\u00ean suy ra $HF \\perp AB$ <br\/> V\u00ec $BE \\perp AD$ (gt) v\u00e0 $BE, HF$ c\u1eaft nhau t\u1ea1i $H$ <br\/> $\\Rightarrow$ $H$ l\u00e0 tr\u1ef1c t\u00e2m tam gi\u00e1c $ABF$ (t\u00ednh ch\u1ea5t ba \u0111\u01b0\u1eddng cao trong tam gi\u00e1c) <br\/> $\\Rightarrow$ $AH \\perp BF$ (t\u00ednh ch\u1ea5t ba \u0111\u01b0\u1eddng cao trong tam gi\u00e1c) (1) <br\/> $\\blacktriangleright$ T\u1eeb $HF = \\dfrac{1}{2}BD $ (cmt) v\u00e0 $AC = \\dfrac{1}{2}BD$ (gt) $\\Rightarrow$ $HF = AC$ <br\/> V\u00ec $HF \/\/ BD$ v\u00e0 $AC \/\/ BD$(v\u00ec c\u00f9ng vu\u00f4ng g\u00f3c v\u1edbi $AB$) <br\/> $\\Rightarrow$ $HF \/\/ AC$ (t\u00ednh ch\u1ea5t) <br\/> $\\Rightarrow$ $\\widehat{CAF} = \\widehat{HFA}$ (hai g\u00f3c so le trong) <br\/> Do \u0111\u00f3, $\\triangle{ACF} = \\triangle{FHA}$ (c.g.c) <br\/> $\\Rightarrow$ $\\widehat{CFA} = \\widehat{HAF}$ (hai g\u00f3c t\u01b0\u01a1ng \u1ee9ng). <br\/> M\u00e0 hai g\u00f3c n\u00e0y \u1edf v\u1ecb tr\u00ed so le n\u00ean $CF \/\/ AH$ (2) <br\/> T\u1eeb (1) v\u00e0 (2) $\\Rightarrow$ $BF \\perp CF$ <br\/> V\u1eady kh\u1eb3ng \u0111\u1ecbnh tr\u00ean l\u00e0 \u0110\u00daNG <\/span> ","column":2}]}]},"correct":"","level":"3","hint":"","answer":"","type":"json","extra_type":"","time":"0","user_id":"0","test":"0","date":"2019-09-30 09:23:07"}]}