Ôn tập HH on tap hh 4681

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BÀI TẬP ÔN TẬP HÌNH HỌC
HỌC KỲ II – NĂM HỌC 2013-2014
Bài 1: Cho tam giác ABC có 3 góc nhọn, hai đường cao BE, CF cắt nhau tại H.
a) CM: AH ⊥ BC.
b) Chứng tỏ: AE.AC = AF.AB
c) Chứng minh: AEF đồng dạng ABC
d) Gọi D là giao điểm của AH với BC. Chứng minh: AEF đồng dạng CED từ đó suy ra: Tia EH là tia
phân giác của góc FED.
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Bài 2: ABC có AB < AC, hai đường cao BD và CE.
a) Chứng minh: ABD đồng dạng ACE. Suy ra AB.AE = AC.AD
b) Chứng minh: ADE đồng dạng ABC.
c) Tia DE và CB cắt nhau tại I. Chứng minh: IBE đồng dạng IDC.
2
2

d) Gọi O là trung điểm của BC.Chứng minh: ID.IE = OI − OC
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Bài 3: : Cho ∆ ABC vuông tại A có AB = 8cm, AC = 6cm, AH là đường cao, AD là đường phân giác.
a) Tính BD và CD
b) Kẻ HE ⊥ AB tại E, HF ⊥ AC tại F. Chứng minh: AE.AB = AH2
c) Chứng minh AE.AB = AF.AC
d) Tính BE.
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Bài 4: Cho tam giác ABC, đường cao AD, BE, CF cắt nhau tại H.
a) Chứng minh: ∆ ABD đồng dạng ∆ CBF .
b) Chứng minh: AH.HD = CH.HF
c) Chứng minh: ∆ BDF đồng dạng ∆ ABC.
d) Gọi K là giao điểm của DE và CF. Chứng minh: HF.CK = HK.CF
Bài 5: ABC (AB < AC) có ba đường cao AD, BE, CF cắt nhau tại H.
a) CM: ∆ AFH đồng dạng ∆ ADB.
b) CM: BH.HE = CH.HF
c) CM: ∆ AEF đồng dạng ∆ ABC.
d) Gọi I là trung điểm của BC, qua H kẻ đường thẳng vuông góc với HI, đường thẳng này cắt đường thẳng
AB tại M và cắt đường thẳng AC tại N. Chứng minh: MH = HN.
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Bài 6: Cho tam giác ABC (AB < AC) có ba góc nhọn, các đường cao AD, BE, CF cắt nhau tại H.

a) Chứng minh: ∆ CFB đồng dạng ∆ ADB.
b) Chứng minh: AF.AB = AH.AD.
c) Chứng minh: ∆ BDF đồng dạng ∆ BAC.
d) Gọi M là trung điểm của BC. Chứng minh: Góc EDF bằng góc EMF
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AB ; F

∈

AC )

b) Chứng minh: AE.AB = AH2 và AE.AB = AF. AC
c) Chứng minh:

∆

dồng dạng ABC .
∆

d) Đường thẳng EF cắt đường thẳng BC tại M. Chứng tỏ rằng: MB.MC = ME.MF
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Bài 9: ABC vuông tại A có AB = 6cm, AC = 8cm.
a) Tính BC.
b) Vẽ đường cao AH của tam giác ABC. Chứng minh: ∆ HAB đồng dạng ∆ HCA
c) Trên BC lấy điểm E sao cho CE = 4cm. Chứng minh: BE2 = BH.BC
d) Tia phân giác của góc ABC cắt AC tại D. Tính SCED
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Bài 10: Cho ∆ ABC vuông tại A, đường cao AH.
a) Chứng minh: ∆ AHB đồng dạng ∆ CHA.
b) Kẻ đường phân giác AD của ∆ CHA và đường phân giác BK của ∆ ABC (D∈BC; K∈AC). BK cắt lần
lượt AH và AD tại E và F. Chứng minh: ∆ AEF đồng dạng ∆ BEH .
c) Chứng minh: KD // AH.

EH KD
=
d) Chứng minh: AB BC
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Bài 11: Cho hình vuông ABCD có M thuộc AB. Gọi N là giao điểm của DM và BC. Qua D kẻ Dx vuông góc với
DN và Dx cắt BC tại K.
a) Chứng tỏ rằng AM.BN = AD.MB
b) Chứng minh tam giác DMK vuông cân.
1
1
+
2
DN 2 không đổi.
c) Chứng minh DK
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Bài 12: Cho ΔABC vuông tại A có AB = 15cm ; BC = 25cm , AH là đường cao (H thuộc BC), BM là phân giác
của góc ABC (M thuộc AC).
a) Tính độ dài AC, AH.
b) Chứng minh: AB2 = BH.BC
NH MA
=
c) Gọi N là giao điểm của BM và AH. Chứng minh: NA MC

d) Tính diện tích tam giác ABN
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Bài 13: Tam giác ABC vuông tại A, AB = a, AC = 3a. Trên cạnh AC lấy các điểm D và E sao cho AD = DE =
EC.
DB DE
;
a) Tính các tỉ số DC DB
b) Chứng minh các tam giác BDE và CDB đồng dạng
·
·
c) Tính tổng AEB + ACB
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Bài 14: Cho tam giác ABC, các đường trung tuyến BD và CE cắt nhau tại G. Qua điểm O thuộc cạnh BC, vẽ OM
song song với CE, ON song song với BD (M ∈ AB, N ∈ AC). MN cắt BD và CE tại I và K.
MH
a) Gọi H là giao điểm của OM với BD. Tính tỉ số MO
1
MI = MN
3
b) Chứng minh rằng
c) Chứng minh rằng: MI = IK = KN

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Bài 15: Cho tam giác ABC có trực tâm H. Gọi M, N theo thứ tự là trung điểm của BC, AC. Gọi O là giao điểm
các đường trung trực của tam giác.
a) CMR: ∆OMN đồng dạng ∆HAB . Tìm tỉ số đồng dạng
b) So sánh độ dài AH và OM

c) Gọi G là trọng tâm tam giác ABC. CMR ∆HAG đồng dạng ∆OMG
d) Chứng minh ba điểm H, G, O thẳng hàng và GH = 2 GO
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Bài 16. Cho tam giác ABC có độ dài hai cạnh là AB = 15 cm, AC = 18 cm. Trên các cạnh AB, AC lần lượt lấy các
điểm D và E sao cho AD = 12 cm, AE = 10 cm.
a) Chứng minh ∆ABC đồng dạng ∆AED
DE MD
MD
=
b) Gọi M, N lần lượt là trung điểm của các cạnh DE và BC. CMR BC NC . Từ đó tính tỉ số NC

·
c) Kẻ phân giác của BAC cắt MN tại K. Chứng minh BC.MK = DE.NK
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Bài 17. Cho hình chữ nhật ABCD có các cạnh AB = 4 cm, BC = 3 cm.
a) Tính độ dài đoạn BD
b) Qua B, vẽ đường thẳng vuông góc với BD cắt đường thẳng CD tại E. Vẽ CF vuông góc với BE tại F.
Chứng minh ∆BCD đồng dạng ∆CFB . Từ đó suy ra độ dài đoạn CF.
c) Gọi O là giao điểm của Ac và BD. Nối EO cắt CF tại I cắt BC tại K. Chứng minh I là trung điểm của

CF.
d) Chứng minh ba điểm D, K, F thẳng hàng.
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Bài 18. Cho hình chữ nhật ABCD (AD < AB). Vẽ AH vuông góc với BD tại H.
a)
b)
c)

d)

Chứng minh ∆HAD đồng dạng ∆ABD
Với AB = 20 cm, AD = 15 cm. Tính độ dài các đoạn thẳng: BD và AH
Chứng minh AH2 = HD.HB
Trên tia đối của tia DA lấy điểm E sao cho DE < AD. Vẽ EM vuông góc với BD tại M, EM cắt AB tại
O. Vẽ AK vuông góc với BE tại K, vẽ AF vuông góc với OD tại F. Chứng minh H, F, K thẳng hàng.

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0
µ
Bài 19. Cho tam giác ABC có A = 90 , AB = 30 cm, AC = 40 cm; đường cao AE; BD là phân giác; F là giao điểm

của AE và BD.

a) ∆ABC đồng dạng ∆EAC . Tính AE
b) Chứng minh BD.EF = BF. AD
c) Chứng minh AF = AD
d) Tính AF
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Bài 20. Cho tam giác ABC vuông tại A có AB = 6 cm, AC = 8 cm, AH là đường cao.
a)
b)
c)

d)

Tính độ dài cạnh BC
Chứng minh hai tam giác HAB và HCA đồng dạng
Trên cạnh BC lấy điểm E sao cho CE = 4 cm. Chứng minh rằng BE2= BH.BC
Tia phân giác của góc ABC cắt cạnh AC tại D. Tính diện tích tam giác CED

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Bài 21. Cho tam giác ABC nhọn, hai đường cao BE và CD.
a) Chứng minh rằng AD.AB = AE.AC
b) Chứng minh hai tam giác ∆ADE và ∆ACB đồng dạng

c) Cho EB = EC, F là trung điểm của EC. Đường thẳng vuông góc với BF tại O vẽ từ E cắt đường thẳng

vuông góc với EC vẽ từ C tại K. Chứng minh rằng EF = CK
d) Chứng minh rằng 5SCFOK = 4SCEK
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Bài 22. Cho tam giác ABC vuông tại B (C ≠ 300). Gọi E, F lần lượt là trung điểm của BC và AC. Đường phân
giác góc BAC cắt EF tại I và cắt BC tại K.
a) Chứng minh tam giác ABK và tam giác IEK đồng dạng

KC AC
=
IE
b) Chứng minh KE

c) Qua K kẻ KH vuông góc với AC tại H. Chứng minh tam giác BKH và tam giác AFI đồng dạng
d) Chứng minh SABC = SABIH
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1
Bài 23: Cho hình thang ABCD ( AB // CD ) có AB = AD = 2 CD . Gọi M là trung điểm của CD. Goi H là giao
điểm của AM và BD .
a) Chứng minh tứ giác ABMD là hình thoi
b) Chứng minh DB vuông góc BC
c) Chứng minh hai tam giác: ∆ADH và ∆CDB đồng dạng
d) Biết AB = 2,5 cm ; BD = 4cm . Tính độ dài BC và diện tích hình thang ABCD.
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Bài 24: Cho hình bình hành ABCD (AB > BC). Lấy điểm M tuỳ ý trên cạnh AB (M ≠ A , M ≠ B). Đường thẳng
DM cắt AC tại K và cắt đường thẳng BC tại N.
a) Chứng minh: ∆ADK đồng dạng với ∆CNK
S KCD
b) Cho AB = 10cm, AM = 6cm. Tính tỉ số diện tích S KAM
c) Chứng minh: KD2 = KM.KN
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Bài 25: Cho hình bình hành ABCD với AC là đường chéo lớn . Vẽ AM ⊥ BC tại M và AN ⊥ CD tại N
a. Chứng minh hai tam giác ABM và AND đồng dạng.
b. Chứng minh: AB.MN = AC.AM
c. Cho AM = 16 cm; AN = 20 cm chu vi của hình bình hành bằng 108 cm. Tính diện tích của hình bình
hành ABCD
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Bài 26: Cho tam giác ABC vuông tại A, BC = 2AB. Gọi M là trung điểm của BC, lấy D đối xứng của A qua M.
a. Chứng minh tứ giác ABCD là hình chữ nhật.
b. Kẻ BE vuông góc với AD và MN vuông góc với AC, BE cắt AC và MN tại P và F. Chứng minh AE.AM
= AP.AN.
c. Chứng minh tứ giác AMCF là hình thoi. Tính diện tích AMCF nếu AB = 10cm.
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Bài 27: Cho hình bình hành ABCD (AC > BD). Kẻ BE, DF vuông góc với AC ( E; F ∈ AC ).
1. Chứng minh: ∆ABE = ∆CDF ; Tứ giác BEDF là hình bình hành.
2. Gọi H và K thứ tự là hình chiếu của C lên AB và AD. Chứng minh: . ∆ADF ∽ ∆ACK .
2
3. Chứng minh: AC = AB.AH + AD.AK .
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Bài 28: Cho hình vuông ABCD, trên cạnh AB lấy điểm E. Kẻ AH ⊥ DE (H ∈ DE) .

a/ Chứng minh: ∆AHE ∽ ∆DAE .

·
·
b/ Chứng minh: DAH = EDC .
·
c/ Trên cạnh AD lấy điểm F sao cho AF = AE. Tính FHC .
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Bài 29: Cho hình chữ nhật ABCD có AB = 3 cm; BC = 4cm. Gọi H là chân đường vuông góc kẻ từ A xuống BD.
1/ Chứng minh: ∆ABH ∽ ∆ACD .
2/ Tính độ dài đoạn thẳng DH.
3/ Gọi M; N theo thứ tự là các điểm thuộc các đoạn BH và CD sao cho
minh: ∆ABM ∽ ∆ACN . Từ đó suy ra AM ⊥ MN .

BM =

1
1
MH; CN = CD
2
3
. Chứng

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AD 1

=
Bài 30: Cho tam giác ABC vuông ở A, có AB = 6cm; AC = 9cm. Trên cạnh AB lấy một điểm D sao cho DB 2 .
Từ D kẻ đường thẳng song song với BC cắt cạnh AC ở E.
a. Tính độ dài đoạn thẳng AD và AE.
b. Tính diện tích tứ giác BDEC.
c. BE cắt CD ở O. Chứng minh tia AO đi qua trung điểm của đoạn BC.
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Bài 31: Cho hình chữ nhật ABCD có ABD = 30 và BD = 10cm. Gọi M và N thứ tự là trung điểm của AB và CD;
E là điểm bất kì thuộc tia đối của tia CB; BD cắt EN và MN thứ tự tại F và O.
1/ Tính độ dài AD và diện tích hình vuông có cạnh là AB.

2/ Chứng minh O là trung điểm của MN.
·
3/ Chứng minh MN là tia phân giác của EMF .

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Bài 32: Cho tam giác ABC có ba góc nhọn. Đường cao AF, BE cắt nhau tại H. Từ A kẻ tia Ax vuông góc với AC,
từ B kẻ tia By vuông góc với BC. Tia Ax và By cắt nhau tại K.
a) Tứ giác AHBK là hình gì? Tại sao?
b) Chứng minh: ∆HAE đồng dạng với ∆HBF.
c) Chứng minh: CE . CA = CF . CB
d) ∆ABC cần thêm điều kiện gì để tứ giác AHBK là hình thoi.
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Bài 33: Cho tam giác ABC, AB = 4cm, AC = 5cm. Từ trung điểm M của AB vẽ một tia Mx cắt AC tại N sao cho
gócAMN = gócACB.
a) Chứng minh: ∆ABC đồng dạng với ∆ANM.
b) Tính NC.
c) Từ C kẻ một đường thẳng song song với AB cắt MN tại K. Tính tỉ số
.
MN
MK
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Bài 34: Cho tam giác vuông ABC (gócA = 90o), đường cao AH. Biết BH = 4cm, CH = 9cm.
a) Chứng minh: AB2 = BH . BC
b) Tính AB, AC.
c) Đường phân giác BD cắt AH tại E (D ∈ AC). Tính
và chứng minh:
.
EA DC
S EBH
=
EH DA
S DBA
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Bài 35. Cho hình bình hành ABCD. Trên cạnh BC lấy điểm F. Tia AF cắt BD và DC lần lượt ở E và G. Chứng
minh:
a) ∆BEF đồng dạng với ∆DEA, ∆DGE đồng dạng với ∆BAE.
b) AE2 = EF . EG
c) BF . DG không đổi khi F thay đổi trên cạnh BC.

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Bài 36. Cho ∆ABC vng tại A , có AB = 6cm , AC = 8cm. Đường phân giác của góc ABC cắt cạnh AC tại D.
Từ C kẻ CE ⊥ BD tại E.
AD
a) Tính độ dài BC và tỉ số DC .
b) CM: ∆ABD ~ ∆EBC. Từ đó suy ra BD.EC = AD.BC
CD CE
=
c) CM: BC BE
d) Gọi EH là đường cao của ∆EBC. CM: CH.CB = ED.EB.
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Ôn tập HH on tap hh 4681

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