\(AB=\sqrt{\left(5-1\right)^2+\left(-3+1\right)^2}=2\sqrt{5}\)

\(AC=\sqrt{\left(0-1\right)^2+\left(1+1\right)^2}=\sqrt{5}\)

\(BC=\sqrt{\left(0-5\right)^2+\left(1+3\right)^2}=\sqrt{29}\)

=>C=3 căn 5+căn 29

\(AB=\sqrt{\left(0+1\right)^2+\left(2+3\right)^2}=\sqrt{26}\)

\(AC=\sqrt{\left(2+1\right)^2+\left(1+3\right)^2}=\sqrt{3^2+4^2}=5\)

\(BC=\sqrt{\left(2-0\right)^2+\left(1-2\right)^2}=\sqrt{5}\)

=>\(C=\sqrt{26}+5+\sqrt{5}\left(cm\right)\)

\(\overrightarrow{AB}=\left(1;-2\right)\Rightarrow AB=\sqrt{5}\)

\(\overrightarrow{AC}=\left(-2;2\right)\Rightarrow AC=2\sqrt{2}\)

\(BC=\left(-3;4\right)\Rightarrow BC=5\)

Chu vi tam giác ABC: \(AB+AC+BC=\sqrt{5}+2\sqrt{2}+5\)

A B → = 3 ; 12 ,   A C → = 4 ; − 1 ⇒ ( A B )   ⃗ . ( A C )   ⃗ = 3 . 4 + 12 . ( - 1 ) = 0   ⇒ ∆ A B C vuông tại A. Trực tâm của tam giác là đỉnh A. Chọn B

\(\left\{{}\begin{matrix}\overrightarrow{BA}=\left(3;-1\right)\\\overrightarrow{BC}=\left(-4;-2\right)\end{matrix}\right.\)

\(\Rightarrow cos\widehat{ABC}=cos\left(\overrightarrow{BA};\overrightarrow{BC}\right)=\dfrac{3.\left(-4\right)+1.2}{\sqrt{3^2+1^2}.\sqrt{\left(-4\right)^2+\left(-2\right)^2}}=-\dfrac{\sqrt{2}}{2}\)

\(\Rightarrow\widehat{ABC}=135^0\)

\(\overrightarrow{AB}=\left(4;-3\right)\Rightarrow AB=5\)

\(\overrightarrow{AC}=\left(6;0\right)\Rightarrow AC=6\)

\(\overrightarrow{BC}=\left(2;3\right)\Rightarrow BC=\sqrt{13}\)

Chu vi tam giác: \(AB+AC+BC=11+\sqrt{13}\)

A(5;5); B(2;1); C(1;2)

\(AB=\sqrt{\left(2-5\right)^2+\left(1-5\right)^2}=\sqrt{\left(-3\right)^2+\left(-4\right)^2}=5\)

\(AC=\sqrt{\left(1-5\right)^2+\left(2-5\right)^2}=5\)

\(BC=\sqrt{\left(1-2\right)^2+\left(2-1\right)^2}=\sqrt2\)

Xét ΔABC có cos BAC=\(\frac{AB^2+AC^2-BC^2}{2\cdot AB\cdot AC}=\frac{25+25-2}{2\cdot5\cdot5}=\frac{50-2}{2\cdot25}=\frac{48}{50}=\frac{24}{25}\)

=>\(\sin BAC=\sqrt{1-\left(\frac{24}{25}\right)^2}=\sqrt{\left(1-\frac{24}{25}\right)\left(1+\frac{24}{25}\right)}=\sqrt{\frac{1}{25}\cdot\frac{49}{25}}=\frac{7}{25}\)

\(S_{BAC}=\frac12\cdot AB\cdot AC\cdot\sin BAC\)

\(=\frac12\cdot5\cdot5\cdot\frac{7}{25}=\frac72=3,5\)

\(\left\{{}\begin{matrix}\overrightarrow{AB}=\left(-1;-2\right)\\\overrightarrow{AC}=\left(4;-2\right)\end{matrix}\right.\)

\(\Rightarrow\overrightarrow{AB}.\overrightarrow{AC}=-1.4+\left(-2\right).\left(-2\right)=0\)

\(\Rightarrow\Delta ABC\) vuông tại A

Ta có B A → = 3 ; − 1  và B C → = − 4 ; − 2 . Suy ra:

cos B A → , B C → = B A → . B C → B A → . B C → = 3. − 4 + − 1 . − 2 9 + 1 . 16 + 4 = − 2 2 ⇒ B ^ = B A → , B C → = 135 O .  

Chọn D.