1.cho tam giác ABC vuông tại A đường cao AH.a)AH nhânBC=AB nhân AC.b)1/AH^2=1/AB^2=1/AC^2
2. Cho tam giác ABC có BD và CE gọi I , K là trung điểm của BC và DE. chứng minh Ik vuông góc với DE.
Câu 1: Cho tam giác ABC vuông tại A, đường cao AH, D và E là 2 đường vuông góc kẻ từ H đến AB và AC.
A) Chứng minh AH=DE
B) I là trung điểm HB, K là trung điểm HC. Chứng minh DI song song với EK
Câu 2: Cho tam giác ABC vuông góc tại A, đường cao AH, trung tuyến AM.
A) Chứng minh góc HAB = góc MAC
B) Vẽ HD vuông góc với AB, HE vuông góc với AC. Chứng minh AM vuông góc với DE.
1a) A=D=E=90 độ
=>AEHD là hcn
=>AH=DE
b)Xét tam giác DBH vuông tại D có:
DI là đường trung tuyến ứng với cạnh huyền BH
=>DI=BH/2=IH
=>tam giác IDH cân tại I
=>góc IDH=góc IHD (1)
Gọi O là gđ 2 đường chéo AH và DE
=>OD=OA=OE=OH (tự c/m)
=> tam giác DOH cân tại O
=> góc ODH=góc OHD(2)
từ (1) và (2) => góc ODH+góc IDH=90 độ(EHD+DHI=90 độ)
=>IDvuông góc DE(3)
Cmtt ta được: KEvuông góc DE(4)
Từ (3)và (4) => DI//KE.
2a) Ta có góc HAB+góc HAC=90 độ (1)
Xét tam giác ABC vuông tại A có
AM là đg trung tuyến ứng vs cạnh huyền BC
=>AM=MC
=>tam giác AMC cân
=>góc MAC=góc ACM
Lại có: góc HAC+góc ACH=90 độ(2)
Từ (1) và (2) => góc BAH=góc ACM
Mà góc AMC=góc MAC(cmt)
=>ABH=MAC(3)
b)A=D=E=90 độ
=>AFHE là hcn
Gọi O là gđ EF và AM
OA=OF(tự cm đi nha)
=>tam giác OAF cân
=>OAF=OFA(4)
Ta có : OAF+MCA=90 độ(5)
Từ (3)(4) và (5)
=>MAC+OFA=90 độ
Hay AM vuông góc EF
k giùm mình nha.
Hình bạn tự kẻ nhá
a) Xét Δ ABC vuông tại A có :
AM là đường trung tuyến
=> AM=1/2BC (tính chất đường trung tuyến trong Δ vuông)
=> AM=MC
=>Δ AMC cân tại M => góc MAC= góc MCA
Mà góc AMC+ Góc ABC = 90° (vì tam giác ABC vuông tại A)
=> góc ABC+ góc MAC = 90° (1)
Xét tam giac vuông AHB có: góc HAB + góc ABC = 90° (2)
Từ (1) và (2) => góc BAH = góc MAC ( cùng phụ với góc ABC )
Vậy góc BAH = góc MAC
1. Cho tam giác ABC vuông ở A có AB<AC. AH vuông góc với BC tại H, D là điểm trên cạnh BC sao cho AD=AB. Vẽ DE vuông góc với BC tại E. Chứng mih rằng AH=HE.
2. Cho tam giác ABC vuông cân tại A.. Qua A vẽ đường thẳng d ở ngoài tam giác ABC . Vẽ BD vuông góc với d taị D. CE vuông góc với d tại E. M là trung điểm CB. Chứng minh rằng:
a) BD + CE = DE
b) Tam giác MDE là tam giác vuông cân
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Bài 1: Cho tam giác ABC vuông tại A có đường phân giác BD, đường trung tuyến AM, đường cao AH.
a) Tính AB, BC, AH, AM. Biết AD = 3 cm; CD = 5 cm.
b) Gọi I, K lần lượt là hình chiếu của H trên AB, AC. Chứng minh rằng AM vuông góc vs IK.
a) Xét ΔABC có BD là đường phân giác ứng với cạnh AC(Gt)
nên \(\dfrac{AD}{AB}=\dfrac{CD}{CB}\)(Tính chất đường phân giác của tam giác)
\(\Leftrightarrow\dfrac{AB}{3}=\dfrac{BC}{5}\)
Ta có: AD+CD=AC(D nằm giữa A và C)
nên AC=3+5=8(cm)
Đặt \(\dfrac{AB}{3}=\dfrac{BC}{5}=k\)
\(\Leftrightarrow\left\{{}\begin{matrix}AB=3k\\BC=5k\end{matrix}\right.\)
Áp dụng định lí Pytago vào ΔABC vuông tại A, ta được:
\(BC^2=AB^2+AC^2\)
\(\Leftrightarrow\left(3k\right)^2+8^2=\left(5k\right)^2\)
\(\Leftrightarrow9k^2+64=25k^2\)
\(\Leftrightarrow16k^2=64\)
\(\Leftrightarrow k^2=4\)
hay k=2
Suy ra: \(\left\{{}\begin{matrix}AB=3\cdot k=3\cdot2=6\left(cm\right)\\BC=5\cdot k=5\cdot2=10\left(cm\right)\end{matrix}\right.\)
Vậy: AB=6cm; BC=10cm
Bài 1:Cho tam giác ABC vuông góc tại A, đường cao AH, phân giác AI của góc HAC
a) Chứng minh tam giác ABI là tam giác cân
b) Cho AB=7,5cm, AC=10 cm. Tính BC,AH,HI,HC,IC
c) Phân giác BE của góc ABC cắt AH tại K(E thuộc AC). Chứng minh IK//AC và tính AE,BE
d) Gọi P là trung điểm của AB. Chứng minh CP đi qua trung điểm của đường vuông góc hạ từ H tới AC
Bài 2: Cho tam giác ABC vuông góc tại C. Gọi M lá trung điểm của AB. Kẻ MD vuông góc CA, ME vuông góc CB
a) Tứ giác CDME là hình gì ? Vì sao ?
b) Gỉa sử AC =5cm, CB =12cm. Tính DE
c) Kẻ đường cao CH của tam giác ABC. Tính CH nếu AC =12cm, AB=15cm
d) Chứng minh CH vuông góc DE
Bài 1:
Cho tam giác ABC cân có AB=AC=5cm, BC= 8cm.Kẻ AH vuông góc với BC ( H thuộc BC).
a, Chứng minh HB=HC
b, Tính độ dài AH.
c, Kẻ HD vuông góc với AB(D thuộc AB), kẻ HE vuông góc với AC ( E thuộc AC).Chứng minh tam giác HDE cân.
d, So sánh HD và HC.
Bài 2:
Cho tam giác ABC cân tại A có đường cao AH.
a, Chứng minh tam giác ABH = tam giác ACH và AH là tia phân giác của góc BAC.
b, Cho BH= 8cm, AB= 10cm.Tính AH.
c,, Gọi E là trung điểm của AC và G là giao điểm của BE và AH.Tính HG.
d, Vẽ Hx song song với AC, Hx cắt AB tại F. Chứng minh C, G, F thẳng hàng.
Bài 3
Cho tam giác ABC có CA= CB= 10cm, AB= 12cm.kẻ CI vuông góc với AB.Kẻ IH vuông góc với AC, IK vuông góc với BC.
a, Chứng minh IB= IC và tính độ dài CI
b, Chứng minh IH= IK.
c, HK// AC.
Bài 4:
Cho tam giác ABC cân tại A, vẽ AH vuông góc với BC tại H.Biết AB= 10cm, BH= 6cm.
a, Tính AH
b, tam giác ABH= tam giác ACH.
c, trên BA lấy D, CA lấy E sao cho BD= CE.Chứng minh tam giác HDE cân.
d, AH là trung trực của DE.
Bài 5:
Cho tam giác ABC cân tại AGọi D là trung điểm của BC.Từ D kẻ DE vuông góc với AB, DF vuông góc với AC. Chứng minh rằng:
a, tam giác ABD= tam giác ACD.
b, AD vuông góc với BC.
c, Cho AC= 10cm, BC= 12cm.Tính AD.
d, tam giác DEF cân.
Bài 6:
Cho tam giác ABC cân tại A có góc A < 900. kẻ BH vuông góc với AC ,CK vuông góc với AC.Gọi O là giao điểm của BH và CK.
a, Chứng minh tam giác ABH=Tam giác ACH.
b, Tam giác OBC cân.
c, Tam giác OBK = tam giác OCK.
d, trên nửa mặt phẳng bờ BC không chứa điểm A lấy I sao cho IB=IC.Chứng minh 3 điểm A, O, I thẳng hàng.
Bài 7
Cho tam giác ABC cân tại A. Kẻ BD vuông góc với AC, CE vuông góc với AB. BD và CE cắt nhau tại H.
a, Tam giác ABD=tam giác ACE.
b, Tam giác BHC cân.
c, ED//BC
d, AH cắt BC tại K, trên HK lấy M sao cho K là trung điểm của HM.Chứng minh tam giác ACM vuông.
Bài 8
Cho tam giác ABC cân tại A. Kẻ BD vuông góc với AC, CE vuông góc với AB. BD và CE cắt nhau tại H.
a, BD= CE.
b, Tam giác BHC cân.
c, AH là trung trực của BC
d, Trên tia BD lấy K sao cho D là trung điểm của BK.So sánh góc ECB và góc DKC.
Bài9
Cho tam giác ABC cân tại A.vẽ trung tuyến AM .từ M kẻ ME vuông góc với AB tại E.kẻ MF vuông góc với AC tại F.
a, chứng minh tam giác BEM= tam giác CFM.
b, AM là trung trực vủa EF.
c, từ B kẻ đường thẳng vuông góc với AB tại B, từ C kẻ đường thẳng vuông góc với AC tại C, hai đường này cắt nhau tại D.Chứng minh A,M,D thẳng hàng.
Bài 10
Cho tam giác ABC cân tại AGọi M là trung điểm của AC.Trên tia đối MB lấy D sao cho DM= BM.
a, Chứng minh Tam giác BMC= tam giác DMA.Suy ra AD//BC.
b, tam giác ACD cân.
c. trên tia đối CA lấy E sao cho CA= CE.Chuwngsminh DC đi qua trung điểm I của BE.
Bài 11: Cho tam giác ABC cân tại A (AB = AC ), M là trung điểm của BC. Gọi D là điểm là điểm nằm giữa A và M. Chứng minh rằng:
a) AM là tia phân giác của góc A?
b) (ABD = (ACD.
c) (BCD là tam giác cân ?
Bài 12: Cho tam giác ABC vuông tại A , đường phân giác BD. Kẻ DE vuông góc với BC (E BC). Gọi F là giao điểm của BA và ED.
Giúp mk với các bạn đẹp trai xinh gái ai làm đúng mk tik cho
Sắp hết Tết rùi giúp mk vs
ủa r viết ngần đó thì mất bn tg thek
Má ơi sao nó dài
cho tam giác ABC vuông tại A ,AH là đường cao BD là đường phân giác kẻ DE vuông góc với BC đường thẳng DE cắt AB tại F tính BC và AH chứng minh tam giác EBF đồng dạng với EDC gọi I là giao điểm AH và BD chứng minh AB.BI =BH.BD chứng minh BD vuông góc với CF
Cho tam giác ABC có AB = 3cm; AC = 4cm; BC = 5cm. Kẻ đường cao AH( H thuộc BC).
1) Chứng tỏ tam giác ABC là tam giác vuông.
2) Trên cạnh BC lấy điểm D sao cho BD = BA, trên cạnh AC lấy điểm E sao cho AE = AH. Gọi F là giao điểm của DE và AH. Chứng minh:
a) DE vuông góc với AC.
b) Tam giác ACF là tam giác cân.
c) BC + AH > AC+ AB
Cho tam giác ABC vuông tại A có AB<AC. Trên tia đối của tia AC lấy điểm D sao cho AD=AB. Trên tia đối của tia AB lấy điểm E sao cho AE=AC
a. So sánh các góc của tam giác ABC. Chứng minh BD<BC
b. Chứng minh BC=DE, tam giác ABC vuông cân và BC//CE
c. Kẻ đường cao AH của tam giác ABC, đường cao AH cắt DE tại M. Từ A kẻ đường vuông góc với CM tại K. đường thẳng này cắt BC tại N. Chứng minh rằng MN//AB
1. Cho tam giác ABC , hai đường cao BD, CE. Gọi M, N lần lượt là trung điểm của BC, DE. Chứng minh MN vuông góc với DE?
2. Cho tam giác ABC có góc A <900. Trên nữa mặt phẳng bờ AB có chứa điểm C vẽ AD vuông góc với AB và AD=AB, trên nữa mặt phẳng bờ AC có chứa điểm B vẽ AE vuông góc với AC và AE=AC. Kẻ AH vông góc với ED (H thuộc ED). Chứng minh rằng đường thẳng Ah đi qua trung điểm M của cạnh BC.