Question

determine a area do triangulo formado pelos pontos A(1,3) B(3,5) C(7,1)

Answer 1

A área da matriz, dados seus vértices, é dada por:

$Area~=~\frac{1}{2}.\left|\left|\begin{array}{ccc}x_A&y_A&1\\x_B&y_B&1\\x_C&y_C&1\end{array}\right|\right| \\\\\\\\Area~=~\frac{1}{2}.\left|\left|\begin{array}{ccc}1&3&1\\3&5&1\\7&1&1\end{array}\right|\right|\\\\\\\\Area~=~\frac{1}{2}~.~\left|~(1.5.1~+~3.1.1~+~7.3.1)~-~(1.5.7~+~1.1.1~+~1.3.3)~\right|\\\\\\Area~=~\frac{1}{2}~.~\left|~(5~+~3~+~21)~-~(35~+~1~+~9)~\right|\\\\\\Area~=~\frac{1}{2}~.~\left|~(29)~-~(45)~\right|\\\\\\Area~=~\frac{1}{2}~.~\left|\,-16~\right|$

$Area~=~\frac{1}{2}~.~16\\\\\\\boxed{Area~=~8~unidades~de~area}$

Answer 2

A = $\frac{1}{2}|D|$

A = $\frac{1}{2} |$$\left[\begin{array}{ccc}1&3&7\\3&5&1\\1&1&1\end{array}\right]$|

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1    3    1    1    3 ⇒ repete primeira e segunda coluna

3    5    1   3   5

7    1     1   7    1

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D = (5 + 21+3) - (35 + 1 + 9)

D = 29 - (45)

D = - 16

A = $\frac{1}{2}|D|$

A = $\frac{1}{2} |-16|$

A = $\frac{16}{2}$

A = 8 u.a.