Question

Dadas as coordenadas A(0,1) B(3-3) C(-1,8)
Obs: use determinantes. considere o metrô (m) como unidade de medida
A. Calcule a área do triângulo ABC ?

Answer 1

A área de um triangulo, dadas as coordenadas dos seus três vértices, é dada por:

$Area~=~\dfrac{1}{2}\cdot\left\|\begin{array}{ccc}x_{_A}&y_{_A}&1\\x_{_B}&y_{_B}&1\\x_{_C}&y_{_C}&1\end{array}\right\|$

Substituindo os pontos:

$Area~=~\dfrac{1}{2}\cdot\left\|\begin{array}{ccc}0&1&1\\3&-3&1\\-1&8&1\end{array}\right\|$

$=~\dfrac{1}{2}\cdot\left|(0\cdot(-3)\cdot1+3\cdot8\cdot1+(-1)\cdot1\cdot1)~-~(1\cdot(-3)\cdot(-1)+1\cdot8\cdot0+1\cdot1\cdot3)\right|$

$=~\dfrac{1}{2}\cdot\left|(0~+~24~-~1)~-~(3~+~0~+~3)\right|\\\\\\=~\dfrac{1}{2}\cdot\left|(23)~-~(6)\right|\\\\\\=~\dfrac{1}{2}\cdot\left|23~-~6\right|\\\\\\=~\dfrac{1}{2}\cdot\left|17\right|\\\\\\=~\dfrac{1}{2}\cdot17\\\\\\=~\boxed{\dfrac{17}{2}~m^2~~ou~~8,5~~m^2}$