a. Xét \(\Delta ABD\left(\widehat{BDA}=90^o\right)\) có:
\(\widehat{B_1}+\widehat{A_1}=90^o\left(1\right)\)
Ta có: \(\widehat{A_1}+\widehat{BAC}+\widehat{A_2}=180^o\) (2 góc kề bù)
\(\Rightarrow\widehat{A_1}+\widehat{A_2}=180^o-\widehat{BAC}=180^o-90^o=90^o\left(2\right)\)
Từ (1) và (2) \(\Rightarrow\widehat{B_1}=\widehat{A_2}\)
Xét \(\Delta BAD\) và \(\Delta ACE\left(\widehat{ADB}=\widehat{CEA}=90^o\right)\) có:
AB = AC (gt)
\(\widehat{B_1}=\widehat{A_2}\left(cmt\right)\)
\(\Rightarrow\Delta BAD=\Delta ACE\) (Cạnh huyền - góc nhọn)
\(\Rightarrow AD=CE;BD=AE\) mà \(DE=AD+AE\)
\(\Rightarrow DE=BD+CE\left(đpcm\right)\)
b. Ta có: AB = AC = 3 cm
\(\Rightarrow\Delta ABC\) cân tại A
\(\Rightarrow\widehat{B_2}=\dfrac{180^o-\widehat{BAC}}{2}=\dfrac{180^o-90^o}{2}=45^o\left(3\right)\)
Xét \(\Delta AMB\) và \(\Delta AMC\) có:
AM chung
MB = MC (GT)
AB = AC (GT)
\(\Rightarrow\Delta AMB=\Delta AMC\left(c.c.c\right)\)
\(\Rightarrow\widehat{A_3}=\widehat{A_4}\) mà \(\widehat{A_3}+\widehat{A_4}=90^o\) (\(\Delta\) ABC vuông tại A)
\(\Rightarrow\widehat{A_3}=\widehat{A_4}=45^o\left(4\right)\)
Từ (3) và (4) \(\Rightarrow\widehat{B_2}=\widehat{A_3}=45^o\)
\(\Rightarrow\Delta AMB\) cân tại M
\(\Rightarrow MA=MB\)
Mặt khác, \(\widehat{DBM}=\widehat{B_1}+\widehat{B_2};\widehat{EAM}=\widehat{A_2}+\widehat{A_4}\)
Mà \(\widehat{B_1}=\widehat{A_2}\left(\Delta BAD=\Delta ACE\right);\widehat{B_2}=\widehat{A_4}\left(cmt\right)\)
\(\Rightarrow\widehat{DBM}=\widehat{EAM}\)
Xét \(\Delta BDM\) và \(\Delta AEM\) có:
\(BD=AE\left(\Delta BAD=\Delta AEM\right)\)
\(\widehat{DBM}=\widehat{EAM}\left(cmt\right)\)
\(BM=AM\left(cmt\right)\)
\(\Rightarrow\Delta BDM=\Delta AEM\left(c.g.c\right)\)
\(\Rightarrow MD=ME\left(5\right)\)
Xét \(\Delta AMB\) có: \(\widehat{B_2}+\widehat{A_3}+\widehat{AMB}=180^o\)
\(\Rightarrow\widehat{AMB}=180^o-\widehat{B_2}-\widehat{A_3}=180^o-45^o-45^o=90^o\left(•\right)\)
\(\Rightarrow\widehat{M_1}+\widehat{M_2}=90^o\left(•\right)\)
G
\(\Delta BDM=\Delta AEM\Rightarrow\widehat{M_1}=\widehat{M_3}\left(••\right)\)
Từ (•) và (••) \(\Rightarrow\widehat{M_3}+\widehat{M_2}=90^o\Rightarrow\widehat{DME}=90^o\left(6\right)\)
Từ (5) và (6) \(\Rightarrow\Delta DME\) cân tại M
Chỗ (suy ra góc AMB ... (•) ) mk bỏ cái (•) )