Question

Cho \(\Delta\)ABC có A\(\left(0;1\right)\) , B\(\left(1;-2\right)\) , C\(\left(6;3\right)\)

Chứng minh \(\Delta\)ABC vuông . Tính chu vi và S_ABC

Answer 1

Có\(\overrightarrow{AB}\left(1;-3\right),\overrightarrow{AC}\left(6;2\right),\overrightarrow{BC}\left(5;5\right)\)

\(\left|\overrightarrow{AB}\right|=\sqrt{1^2+\left(-3\right)^2}=\sqrt{10}\)

tương tự \(\left|\overrightarrow{AC}\right|=2\sqrt{10},\left|\overrightarrow{BC}\right|=5\sqrt{2}\)

Có \(AB^2+AC^2=\left(\sqrt{10}\right)^2+\left(2\sqrt{10}\right)^2=50=BC^2\)

\(\Rightarrow\Delta ABC\) là tam giác vuông

\(P_{\Delta ABC}=2\sqrt{10}+\sqrt{10}+5\sqrt{2}=3\sqrt{10}+5\sqrt{2}\)

\(S_{\Delta ABC}=\frac{1}{2}.2\sqrt{10}.\sqrt{10}=10\)