cho tam giác ABC nhọn có BC=a; AC=b; AB=c;CMR: a/sinA=b/sinB=c/sin C
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a) Xét ΔAEB vuông tại E và ΔAFC vuông tại F có
\(\widehat{FAC}\) chung
Do đó: ΔAEB\(\sim\)ΔAFC(g-g)
Suy ra: \(\dfrac{AE}{AF}=\dfrac{AB}{AC}\)(Các cặp cạnh tương ứng tỉ lệ)
hay \(AE\cdot AC=AF\cdot AB\)(ĐPCM)
b)
Ta có: \(\dfrac{AE}{AF}=\dfrac{AB}{AC}\)(cmt)
nên \(\dfrac{AE}{AB}=\dfrac{AF}{AC}\)
Xét ΔAEF và ΔABC có
\(\dfrac{AE}{AB}=\dfrac{AF}{AC}\)(cmt)
\(\widehat{FAE}\) chung
Do đó: ΔAEF\(\sim\)ΔABC(c-g-c)
a: Xét ΔAKB và ΔAKC có
AB=AC
góc BAK=góc CAK
AK chung
=>ΔAKB=ΔAKC
ΔABC cân tại A
mà AK là phân giác
nên AK vuông góc CB
b: Xét ΔACB có
BM,AK là trung tuyến
BM cắt AK tại G
=>G là trọng tâm
c: BK=CK=18/2=9cm
=>\(AK=\sqrt{30^2-9^2}=3\sqrt{91}\left(cm\right)\)
=>\(AG=2\sqrt{91}\left(cm\right)\)
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Từ A kẻ đường cao AH (H thuộc BC) , Từ B kẻ đường cao BK (K thuộc AC)
Ta có : \(sinA=\frac{BK}{AB}\) ; \(sinB=\frac{AH}{AB}\) ; \(sinC=\frac{AH}{AC}\)
\(\Rightarrow\frac{AB}{sinC}=\frac{AB}{\frac{AH}{AC}}=\frac{AB.AC}{AH}\) ; \(\frac{AC}{sinB}=\frac{AC}{\frac{AH}{AB}}=\frac{AB.AC}{AH}\)
\(\Rightarrow\frac{c}{sinC}=\frac{b}{sinB}\) (1)
Lại có : \(BK=sinC.BC\Rightarrow\frac{BC}{sinA}=\frac{BC}{\frac{BK}{AB}}=\frac{BC.AB}{BK}=\frac{AB.BC}{sinC.BC}=\frac{AB}{sinC}\)
\(\Rightarrow\frac{a}{sinA}=\frac{c}{sinC}\) (2)
Từ (1) và (2) ta có : \(\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}\) (Đpcm)
mik k hieu lam!