cho tam giác ABC vuông cân tại A . Vẽ AH vuông góc với BC tại H . Chứng minh AH2=BH.CH
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30 tháng 7 2023
Xét ΔAHB vuông tại H và ΔAHC vuông tại H có
AB=AC
AH chung
=>ΔAHB=ΔAHC
=>góc BAH=góc CAH
Xét ΔAMH vuông tại M và ΔANH vuông tại N có
AH chung
góc MAH=góc NAH
=>ΔAMH=ΔANH
=>NH=MH
AH^2-AN^2=NH^2
BH^2-BM^2=MH^2
mà NH=MH
nên AH^2-AN^2=BH^2-BM^2
=>AH^2+BM^2=AN^2+BH^2
16 tháng 3 2022
Xét \(\Delta AHB\) vuông tại H và \(\Delta AHC\) vuông tại H:
\(AB=AC\) (\(\Delta ABC\) cân tại A).
\(\widehat{B}=\widehat{C}\) (\(\Delta ABC\) cân tại A).
\(\Rightarrow\Delta AHB=\) \(\Delta AHC\left(ch-gn\right).\)
\(\Rightarrow\widehat{BAH}=\widehat{CAH}.\)
Xét \(\Delta AMH\) vuông tại M và \(\Delta ANH\) vuông tại N:
\(AHchung.\\ \widehat{MAH}=\widehat{NAH}\left(\widehat{BAH}=\widehat{CAH}\right).\\ \Rightarrow\Delta AMH=\Delta ANH\left(ch-gn\right).\)
Xét \(\Delta AMN:AM=AN\left(\Delta AMH=\Delta ANH\right).\)
\(\Rightarrow\Delta AMN\) cân tại A.
\(\Rightarrow\widehat{AMN}=\dfrac{180^o-\widehat{A}}{2}.\)
Mà \(\widehat{ABC}=\dfrac{180^o-\widehat{A}}{2}\) (\(\Delta ABC\) cân tại A).
\(\Rightarrow\widehat{AMN}=\widehat{ABC}.\\ \Rightarrow MN//BC.\)
20 tháng 2 2021
a) Xét ΔAHB vuông tại H và ΔAHC vuông tại H có
AB=AC(ΔABC cân tại A)
AH chung
Do đó: ΔAHB=ΔAHC(Cạnh huyền-cạnh góc vuông)
b) Ta có: ΔAHB=ΔAHC(cmt)
nên \(\widehat{BAH}=\widehat{CAH}\)(hai góc tương ứng)
hay \(\widehat{MAH}=\widehat{NAH}\)
Xét ΔMAH vuông tại M và ΔNAH vuông tại N có
AH chung
\(\widehat{MAH}=\widehat{NAH}\)(cmt)
Do đó: ΔMAH=ΔNAH(cạnh huyền-góc nhọn)
Suy ra: AM=AN(hai cạnh tương ứng)
Xét ΔMAN có AM=AN(cmt)
nên ΔAMN cân tại A(Định nghĩa tam giác cân)
27 tháng 2 2022
a: Xét ΔABH vuông tại H và ΔACH vuông tại H có
AB=AC
AH chung
Do đó: ΔABH=ΔACH
Suy ra: \(\widehat{BAH}=\widehat{CAH}\)
hay AH là tia phân giác của góc BAC
b: Xét ΔEAH vuông tại E và ΔFAH vuông tại F có
AH chung
\(\widehat{EAH}=\widehat{FAH}\)
Do đó: ΔEAH=ΔFAH
Suy ra: HE=HF
hay ΔHEF cân tại H
c: Xét ΔACK và ΔABK có
AC=AB
\(\widehat{CAK}=\widehat{BAK}\)
AK chung
Do đó: ΔACK=ΔABK
Suy ra: \(\widehat{ACK}=\widehat{ABK}=90^0\)
=>BK\(\perp\)AB
hay BK//EH
15 tháng 2 2019
Chúc bạn học giỏi!
Chúc bạn học tốt!
Chúc bạn học nhanh!
Chúc bạn học siêu!
15 tháng 3 2022
Câu 4:
a: Xét ΔABH vuông tại H và ΔACH vuông tại H có
AB=AC
AH chung
Do đó: ΔABH=ΔACH
b: Xét ΔAEH vuông tại E và ΔAFH vuông tại F có
AH chung
\(\widehat{EAH}=\widehat{FAH}\)
Do đó: ΔAEH=ΔAFH
Suy ra:HE=HF
CL
13 tháng 2 2016
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CV
7 tháng 3 2017
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCGCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
∆CBA;
29 tháng 7 2023
4:
a: Xét ΔACH vuông tại H và ΔBCA vuông tại A có
góc ACH chung
=>ΔACH đồng dạng với ΔBCA
b: Xét ΔHAB vuông tại H và ΔHCA vuông tại H có
góc HAB=góc HCA
=>ΔHAB đồng dạng với ΔHCA
=>HA/HC=HB/HA
=>HA^2=HB*HC
c: góc EHD=góc EHA+góc DHA
=1/2*góc AHB+1/2*góc AHC=90 độ
góc EAD+góc EHD=180 độ
=>EADH nội tiếp
=>góc AED=góc AHD và góc ADE=góc AHE
mà góc AHD=góc AHE=45 độ
nên góc AED=góc ADE
=>AD=AE
Xét tg vuông ABH có \(AH^2=AB^2-BH^2\)
Xét tg vuông ACH có \(AH^2=AC^2-CH^2\)
\(\Rightarrow2.AH^2=\left(AB^2+AC^2\right)-\left(BH^2+CH^2\right)=\)
\(\Rightarrow2.AH^2=BC^2-\left(BH^2+CH^2\right)\)
\(\Rightarrow2.AH^2=\left(BH+CH\right)^2-\left(BH^2+CH^2\right)\)
\(\Rightarrow2.AH^2=BH^2+CH^2+2.BH.CH-BH^2-CH^2=2.BH.CH\)
\(\Rightarrow AH^2=BH.CH\)