Question

Dado o triângulo ABC, representado a seguir, no plano cartesiano, determine a sua área utilizando o conceito de determinantes:

Answer 1

$\boxed{\begin{array}{l}\sf sejam~A(x_A, y_A), B(x_B, y_B) ~e~C(x_C, y_C)\\\sf os~v\acute ertices~de~um~tri\hat angulo~no~plano~cartesiano.\\\sf ent\tilde ao~ A_{tri\hat angulo}=\dfrac{|det~M|}{2}\\\sf onde\\\sf M=\begin{bmatrix}\sf x_A&\sf y_A&\sf1\\\sf x_B&\sf y_B&\sf1\\\sf x_C&\sf y_C&\sf1\end{bmatrix} \end{array}}$

$\rm os~v\acute ertices~do~tri\hat angulo~s\tilde ao~ A(0,0), B(30,10)~e~C(40,0).\\\tt M=\begin{bmatrix}\sf0&\sf0&\sf1\\\sf30&\sf10&\sf1\\\sf40&\sf0&\sf1\end{bmatrix}\\\sf det~M=0\cdot(10-0)-0\cdot(30-40)+1\cdot(0-400)\\\sf det~M=-400$

$\boxed{\begin{array}{l}\sf A_{tri\hat angulo}=\dfrac{|det~M|}{2}\\\sf A_{tri\hat angulo}=\dfrac{|-400|}{2}\\\sf A_{tri\hat angulo} =\dfrac{400}{2}\\\sf A_{tri\hat angulo}=200~u\cdot a\end{array}}$