Question

Calcule o perimetro do triangulo abc sendo a(2,0),b(-2,1)ec(3,4)

Answer 1

Vamos determinar os valores de cada lado por geometria analítica.

AB = B - A

AB = (-2, 1) - (2, 0)

AB = (-2-2, 1-0)

AB = (-4, 1)

Calcule o módulo

|AB| = √(-4)²+1²

|AB| = √17

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AC = C-A

AC = (3,4) - (2,0)

AC = (3-2, 4-0)

AC = (1, 4)

Calcule o módulo:

|AC| = √1²+4²

|AC| = √17
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BC = C - B

BC = (3,4) - (-2, 1)

BC = (3+2, 4-1)

BC = (5, 3)

Calcule o módulo:

|BC| = √5²+3²

|BC| = √34

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Perímetro será a soma dos lados:

$\\ P = AB + AC + BC \\ \\ P = \sqrt{17} + \sqrt{17} + \sqrt{34} \\ \\ P = 2 \sqrt{17} + \sqrt{2*17} \\ \\ P = 2 \sqrt{17}+ \sqrt{2} * \sqrt{17} \\ \\ P = \sqrt{17}( 2 + \sqrt{2} )$

Answer 2

$d_{AB}=\sqrt{(-2-2)^2+(1-0)^2}=\sqrt{16+1}=\sqrt{17}\\ \\ d_{AC}=\sqrt{(3-2)^2+(4-0)^2}=\sqrt{1+16}=\sqrt{17}\\ \\ d_{BC}=\sqrt{(3-(-2)^2+(4-1)^2)}=\sqrt{25+9}=\sqrt{34}\\ \\ P=\sqrt{17}+\sqrt{17}+\sqrt{34}=2\sqrt{17}+\sqrt{34}$