Vì tam giác ABC vuông tại A nên:   A B → . A C → = 0

A B → . B C → = A B → . A C → − A B → = A B → . A C → − A B → 2 = 0 − A B → 2 = − a 2

Chọn C.

Đáp án D

Áp dụng định lý Py – ta – go ta có:

Vì tam giác ABC vuông tại A nên: A B → . A C → = 0

A C → . B C → = A C → . A C → − A B → = A C → 2 −   A C → . A B → = A C 2 − 0 = a 2

Chọn  B.

\(1,HC=\dfrac{AH^2}{BH}=\dfrac{256}{9}\\ \Rightarrow AB=\sqrt{BH\cdot BC}=\sqrt{\left(\dfrac{256}{9}+9\right)9}=\sqrt{337}\\ 2,BC=\sqrt{AB^2+AC^2}=10\left(cm\right)\\ \Rightarrow BH=\dfrac{AB^2}{BC}=6,4\left(cm\right)\\ 3,AC=\sqrt{BC^2-AB^2}=9\\ \Rightarrow CH=\dfrac{AC^2}{BC}=5,4\\ 4,AC=\sqrt{BC\cdot CH}=\sqrt{9\left(6+9\right)}=3\sqrt{15}\\ 5,AC=\sqrt{BC^2-AB^2}=4\sqrt{7}\left(cm\right)\\ \Rightarrow AH=\dfrac{AB\cdot AC}{BC}=3\sqrt{7}\left(cm\right)\\ 6,AC=\sqrt{BC\cdot CH}=\sqrt{12\left(12+8\right)}=4\sqrt{15}\left(cm\right)\)

ohaan ????

a ) Xét \(\Delta ABC\)và \(\Delta EBD\) ta có :

\(\widehat{BAD}\)\(=\) \(\widehat{BED}\)( \(BD\) là phân giác \(\widehat{ABC}\))

\(BD\) là cạnh chung . 

\(\widehat{ABD}\)\(=\) \(\widehat{EBD}\) \(\left(=90^o\right)\)

\(\Rightarrow\Delta ABC=\Delta EBD\) ( g.c.g ) \(\Rightarrow AD=ED\) và \(AB=EB\)( 1 )

b )  

\(\left(1\right)\)\(\Rightarrow AD=DE\)

Xét \(\Delta HAD\)và \(\Delta EDC\)có: 

\(\widehat{HAD}\)\(=\) \(\widehat{CED}\)\(=\) \(90^o\)

\(AD=DE\) 

\(\widehat{ADH}\)\(=\) \(\widehat{EDC}\) ( đối đỉnh ) 

\(\Rightarrow\Delta ADH=\Delta EDC\) ( g.c.g )  ( 2 )

c,  

\(\left(2\right)\)\(\Rightarrow AH=EC\)

Xét \(\Delta AHC\)và \(\Delta ECH\) có: 

\(\widehat{HAC}\)\(=\) \(\widehat{CEH}\)\(=90^o\) 

\(HC\) là cạnh chung .

\(HA=CE\)

\(\Rightarrow\Delta HAC=\Delta CEH\)  ( ch  .cgv ) 

d,  

\(\left(1\right)\)\(\Rightarrow AB=BE\) 

Xét \(\Delta BEH\) và \(\Delta BAC\) có: 

\(\widehat{BEH}\)\(=\) \(\widehat{BAC}\)\(=90^o\)  

\(BE=AB\)

\(\widehat{HBC}\) chung . 

\(\Rightarrow\Delta BEH=\Delta BAC\)  ( g.c.g )

ta có \(\left(\overrightarrow{AB}+\overrightarrow{AC}\right)^2=AB^2+2\overrightarrow{AB}.\overrightarrow{AC}+AC^2=AB^2+AC^2=5^2+12^2=13^2\)

Vậy \(\left|\overrightarrow{AB}+\overrightarrow{AC}\right|=\sqrt{13^2}=13\)

còn \(\overrightarrow{BA}-\overrightarrow{CB}=\overrightarrow{BA}-\overrightarrow{CA}+\overrightarrow{BA}=2\overrightarrow{BA}-\overrightarrow{CA}\)

Mà \(\left(2\overrightarrow{BA}-\overrightarrow{CA}\right)^2=4BA^2-4\overrightarrow{BA}.\overrightarrow{CA}+CA^2=4BA^2+CA^2=4.5^2+12^2=244\)

vậy \(\left|\overrightarrow{BA}-\overrightarrow{CB}\right|=\sqrt{244}\)

Ta có: tam giác ABC vuông tại B nên  A B → ⊥ ​ B C → ⇒ A B → .   ​ B C → = 0

  A B → .    A C → = A B → .   (   A B → + ​ B C → ) = A B → 2 + ​ A B → .    B C → = A B 2 + 0 = 81

Chọn  C.