Do P thuộc Ox nên tọa độ có dạng \(P\left(p;0\right)\)

\(\Rightarrow\left\{{}\begin{matrix}\overrightarrow{MN}=\left(1;-3\right)\\\overrightarrow{MP}=\left(p-2;-1\right)\end{matrix}\right.\)

Do tam giác MNP vuông tại M \(\Rightarrow\overrightarrow{MN}.\overrightarrow{MP}=0\)

\(\Rightarrow1.\left(p-2\right)+3=0\) \(\Rightarrow p=-1\)

\(\Rightarrow P\left(-1;0\right)\)

\(\Rightarrow\overrightarrow{MP}=\left(-3;-1\right)\Rightarrow\left\{{}\begin{matrix}MN=\sqrt{1^2+\left(-3\right)^2}=\sqrt{10}\\MP=\sqrt{\left(-3\right)^2+\left(-1\right)^2}=\sqrt{10}\end{matrix}\right.\) 

\(\Rightarrow S_{MNP}=\dfrac{1}{2}MN.MP=5\)

Do C thuộc trục tung nên tọa độ có dạng \(C\left(0;c\right)\Rightarrow\left\{{}\begin{matrix}\overrightarrow{AB}=\left(-4;-1\right)\\\overrightarrow{AC}=\left(-1;c-2\right)\end{matrix}\right.\)

Do tam giác ABC vuông tại A \(\Rightarrow\overrightarrow{AB}.\overrightarrow{AC}=0\)

\(\Rightarrow4-\left(c-2\right)=0\Rightarrow c=6\)

\(\Rightarrow C\left(0;6\right)\)

\(\Rightarrow\overrightarrow{AC}=\left(-1;4\right)\Rightarrow\left\{{}\begin{matrix}AB=\sqrt{\left(-4\right)^2+\left(-1\right)^2}=\sqrt{17}\\AC=\sqrt{\left(-1\right)^2+4^2}=\sqrt{17}\end{matrix}\right.\)

\(\Rightarrow S_{ABC}=\dfrac{1}{2}AB.AC=\dfrac{17}{2}\)

Gọi \(H\left(x;y\right)\) là trực tâm tam giác

\(\Rightarrow\overrightarrow{AH}=\left(x+3;y\right)\) ; \(\overrightarrow{BH}=\left(x-3;y\right)\); \(\overrightarrow{BC}=\left(-1;6\right)\) ; \(\overrightarrow{AC}=\left(5;6\right)\)

Do H là trực tâm tam giác \(\Rightarrow\left\{{}\begin{matrix}AH\perp BC\\BH\perp AC\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}\overrightarrow{AH}.\overrightarrow{BC}=0\\\overrightarrow{BH}.\overrightarrow{AC}=0\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}-\left(x+3\right)+6y=0\\5\left(x-3\right)+6y=0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}-x+6y=3\\5x+6y=15\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}x=2\\y=\dfrac{5}{6}\\\end{matrix}\right.\) \(\Rightarrow H\left(2;\dfrac{5}{6}\right)\)

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

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

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

\(C=180^0-\left(A+B\right)=77^04'\)

Áp dụng định lý hàm sin:

\(\dfrac{AB}{sinC}=\dfrac{AC}{sinB}\Rightarrow AC=\dfrac{AB.sinB}{sinC}=\dfrac{117.sin34^044'}{sin77^04'}\approx68,4\)

Áp dụng định lý hàm cosin:

\(AC=\sqrt{AB^2+BC^2-2AB.BC.cosB}=\sqrt{2^2+3^2-2.2.3.cos60^0}=\sqrt{2}\)

Diện tích tam giác:

\(S=\dfrac{1}{2}AB.BC.sinB=\dfrac{1}{2}.2.3.sin60^0=\dfrac{3\sqrt{3}}{2}\)

\(A=180^0-\left(B+C\right)=52^047'\)

Áp dụng định lý hàm sin:

\(\dfrac{a}{sinA}=\dfrac{c}{sinC}\Rightarrow c=\dfrac{a.sinC}{sinA}\approx19,9\)

Ta có: \(\widehat{C}=180^0-\left(\widehat{A}+\widehat{B}\right)=180^0-\left(40^0+60^0\right)=80^0\)

Áp dụng định lý sin vào △ABC có:

\(\dfrac{BC}{\sin A}=\dfrac{AB}{\sin C}\) 

\(\Rightarrow BC=\dfrac{AB.\sin A}{\sin C}=\dfrac{5.\sin40}{\sin60}\approx3,26\)

\(cosA=\dfrac{AB^2+AC^2-BC^2}{2AB.AC}=\dfrac{2}{3}\)

\(sinA=\sqrt{1-cos^2A}=\dfrac{\sqrt{5}}{3}\)

\(\Rightarrow S_{ABC}=\dfrac{1}{2}AB.AC.sinA=6\sqrt{5}\)

Áp dụng định lý hàm cosin:

\(b=\sqrt{a^2+c^2-2ac.cosB}=7\)

Diện tích:

\(S_{ABC}=\dfrac{1}{2}ac.sinB=10\sqrt{3}\)