. Dovay dÚcJng tron (K) tiep xuc trong vdi dtfdng
a)Ttfgiac ABCD la hinh chỉ nhat c6 tarn OA ^ OA = OB = OC = OD.
Ma ABD = 30" nen ADO = 60° =^ AAOD deu => AOD = 60"
AAOD deu, AG la diTcfng trung tuyen ^ nen cung la dúdng cao, dúcJng phan giac.
Ta c6: DAE = ^ D A O = 30", A E 1 B D . Suy ra AED = 60". Ta c6: AED = AOD (= 60") => TuT giac AOED noi tiep diTdc. b) AADF = AODF (c.g.c) => AF = OF, DOT = DAF = 90"
AABD vuong tai A => A D = ABtg ABD = 3ạtg30" = 73 a AADF vuong tai A => AF = ADtg ADF = V3 atg30" = a
V3a
•=2a AADE vuong tai A => A D = AEcos ADE => AE =
cos30"
AE 1 BD, OF 1 BD => AE // OF =^ Ttf gidc AFOE la hinh thang.
Do vay: SAFOE = - ( O F + AE)OT = i( a + 2a) — = (dvdt) 2 2 2 4
Cty TNHH MTV DWH Khang Vỉt
c) ABOJ = ABCJ (c.g.c) (OB = EC, BJ canh chung, OBJ = CBJ) => BOJ = BCJ . Ma BCJ = - B C D = 45". Vay IOJ = 45". => BOJ = BCJ . Ma BCJ = - B C D = 45". Vay IOJ = 45". ADKI can tai D, K D I = 30" ^ D K I = 75".
Ma D K H = KHC + K C H nen KHC = 75" - 30" = 45".
AKJC vuong can tai K KJC = 45" . . . Ta c6: KHC = KJC (= 45") => TiJ gidc KHJC noi tiep => JHC = JKC = 90" BJ 1 OC, JH 1 OC => B, J, H th^ng hang => H la trung diem OC.
Ta c 6: 0 H = i 0 C = i A D = - ^ .
2 2 2
AAOE = AADE (c.g.c) => AOE = ADE = 90" ; OE = DE = AEsinDAE = a
'V3af
AOEH vuong tai O => EH^ = OÊ + O t f => E t f = â +
CO
Vay EH = .
Nhan xet: a) De thay AED = AOD = 60".
b) Cac so" do 30", 60" giup nghl den // soluang gidc ciia goc nhon.
c) Nhan ra IOJ= 45", hai tarn giac BOJ, BCJ b^ng nhau giup c6 difdc dieu do ; B, J, H thing hang giup c6 H la trung diem OC. Tuf do tinh dtfcJc do dai do ; B, J, H thing hang giup c6 H la trung diem OC. Tuf do tinh dtfcJc do dai doan HE theo ạ
O i THI TUY^'N SINH VAO LdP 10 CHUY^N, '
§ TRl/dNG PHd THONG NANG KHIEU, DAI HQC QUÓC GIA, TP.HCM
NAM HOC 2010 - 2011
B & i l: ( 2 d i e m ) '
a) Cho a, b, c la cdc so thiTc th6a man dieu kien: a + b + c = á + b ' + c' = 0. ChiJng minh r^ng trong ba so a, b, c CO It nhát mot so bkng 0. ChiJng minh r^ng trong ba so a, b, c CO It nhát mot so bkng 0.
'x + y + z = 3 . X 4 •
xy + yz + zx = - 1 . , , x^ + y3 + + 6 = 3(x^ + + z^) , :
Bai 2: (2 diem) , . < < > • a) Giai phiTdng tnnh: (2x - 1)^ = 12\/x^ - x - 2 + 1. ; b) Giai he phUdng trinh: (adsbygoogle = window.adsbygoogle || []).push({});
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b) Cho tam giac ABC vuong tai A va c6 dien tich bang 1. Chtfng minh rang ta
CO bat dang thu-c: 2 < BC < V2 (AB + AC - V2) ; ^ ,. v j
Bai 3: (2 diem) •... . ^ v--
a) Hay chi ra mot bo 4 so nguyen diTcfng phan biet ma tong ba so baft ki trong chung la mot so nguyen tọ
b) ChiJng minh rKng khong ton tai 5 só nguyen diTcfng phan biet sao cho tong ba so bat ki trong chung la mot so nguyen tọ
Bai 4: (2 diem) Cho dúdng tron tam O, ban kinh R va day cung BC co dinh c6 do dai BC = RN/3 . A la mot diem thay đi tren cung Idn BC. Goi E la diem doi xiJng cua B qua AC va F la diem doi xtfng cija C qua AB. Cac diTdng
tron ngoai tiép cac tam giac ABE va ACFc^t nhau tai K(K^ A).
a) Chtfng minh K luon thuoc mot dúdng tron có dinh.
b) Xac dinh vi tri cua diem A de tam giac KBC co dien tich Idn nhát va tim gia '*' tri Idn nhat do theo R.
c) Gpi H la giao diem cua BE va CF. Chtfng minh tam giac ABH dong dang vdi tam giac AKC va difdng th^ng AK luon di qua mot diem co dinh.
Bai 5: (2 diem) Trong mot giai bong da co 12 doi tham diT, thi dau vong tron
mot liTdt (hai doi bat ki thi dáu vdi nhau dung mot trSn).
a) ChiJug minh r^ng sau 4 vong dáu (moi dpi thi dáu dung 4 tran) luon tim dúdc ba dpi bong doi mot chiTa thi dáu vdi nhaụ
b) Khing dinh tren con dung khong néu moi dpi da thi dáu dung 5 tran ?