Question

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Answer 1

Explicație pas cu pas:

x - y + 2 = 0 => y = x + 2

x + y - 2 = 0 => y = -x + 2

$d_{1} \cap d_{2} \iff x + 2 = - x + 2 \\ 2x = 0 \implies x = 0 \\ y = 2 \implies A(0;2)$

$d_{1} \cap d_{3} \iff x + 2 = - 4 \\ \implies x = 0 \implies B( - 6; - 4)$

$d_{2} \cap d_{3} \iff - x + 2 = - 4 \\ \implies x = 6 \implies C(6; - 4)$

$AB = \sqrt{ {( - 6 - 0)}^{2} + {( - 4 - 2)}^{2} } = \sqrt{ {6}^{2} + {6}^{2} } = \sqrt{72} = \bf 6 \sqrt{2}$

$AC = \sqrt{ {(6 - 0)}^{2} + {( - 4 - 2)}^{2}} = \sqrt{ {6}^{2} + {6}^{2}} = \sqrt{72} = \bf 6 \sqrt{2}$

$BC = \sqrt{ {(6 - ( - 6))}^{2} + {( - 4 - ( - 4))}^{2} } = \sqrt{ {12}^{2} + 0} = \bf 12$

$P(\triangle ABC) = AB + AC + BC = 2 \cdot 6\sqrt{2} + 12 = \bf 12( \sqrt{2} + 1)$