Cho tam giác ABC vuông tại A có \(\widehat{ABC}=60^0\)
1) Tính số đo \(\widehat{ACB}\)
2) Trên tia đối của tia AC lấy điểm D sao cho AD = AC. Chứng minh \(\Delta ABD=\Delta ABC\)
3) Tính số đo \(\widehat{DBC}\)
4) Qua C vẽ đường thẳng vuông góc với AC, đường thẳng này cắt tia phân giác \(\widehat{ABC}\)tại E. Tính \(\widehat{BCE}\) và chứng minh \(\Delta DBC=\Delta ECB\). Hãy suy ra \(AC=\frac{1}{2}BE\)
(Vẽ hình giúp mik nha)
a) Ta có \(\Delta ABC\) vuông tại \(A\left(gt\right).\)
=> \(\widehat{ABC}+\widehat{ACB}=90^0\) (tính chất tam giác vuông)
Mà \(\widehat{ABC}=60^0\left(gt\right)\)
=> \(60^0+\widehat{ACB}=90^0\)
=> \(\widehat{ACB}=90^0-60^0\)
=> \(\widehat{ACB}=30^0.\)
b) Xét 2 \(\Delta\) vuông \(ABD\) và \(ABC\) có:
\(\widehat{BAD}=\widehat{BAC}=90^0\)
\(AD=AC\left(gt\right)\)
Cạnh AB chung
=> \(\Delta ABD=\Delta ABC\) (cạnh huyền - cạnh góc vuông).
c) Gọi \(Bx\) là tia phân giác của \(\widehat{ABC}.\)
=> \(\widehat{ABx}=\widehat{xBC}=\frac{\widehat{ABC}}{2}=\frac{60^0}{2}=30^0.\)
Vì \(AC\perp EC\left(gt\right)\)
=> \(\widehat{ACE}=90^0\)
Hay \(\widehat{xCE}=90^0.\)
Mà![](data:image/png;base64,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)
=> \(30^0+90^0=\widehat{BCE}\)
=> \(\widehat{BCE}=120^0.\)
Vì \(\Delta ABD=\Delta ABC\left(cmt\right)\)
=> \(\widehat{ABD}=\widehat{ABC}=60^0\) (2 góc tương ứng)
Ta có: \(\widehat{ABD}+\widehat{ABC}=\widehat{DBC}\)
=> \(60^0+60^0=\widehat{DBC}\)
=> \(\widehat{DBC}=120^0.\)
d) Theo câu c) ta có \(\left\{{}\begin{matrix}\widehat{ECB}=120^0\\\widehat{DBC}=120^0\end{matrix}\right.\)
=> \(\widehat{DBC}=\widehat{ECB}=120^0.\)
Xét 2 \(\Delta\) \(DBC\) và \(ECB\) có:
\(\widehat{DBC}=\widehat{ECB}\left(cmt\right)\)
\(\widehat{xBC}=\widehat{C_1}=30^0\)
Cạnh BC chung
=> \(\Delta DBC=\Delta ECB\left(g-c-g\right).\)
=> \(CD=EB\) (2 cạnh tương ứng)
Ta có: \(AD=AC\left(gt\right)\)
=> \(A\) là trung điểm của \(CD.\)
=> \(AC=\frac{1}{2}CD\) (tính chất trung điểm)
Mà \(CD=EB\left(cmt\right)\)
=> \(AC=\frac{1}{2}EB\left(đpcm\right).\)
Chúc bạn học tốt!