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Resposta:
[tex]\textsf{Leia abaixo}[/tex]
Explicação passo a passo:
[tex]\sf{A(k,4)}[/tex]
[tex]\sf{r:2x + 8y - 80 = 0}[/tex]
[tex]\mathsf{d_{A,r} = 6}[/tex]
[tex]\mathsf{d_{p,r} = |\:\dfrac{ax_0 + by_0 + c}{\sqrt{a^2 + b^2}}\:|}[/tex]
[tex]\mathsf{6 = |\:\dfrac{2k + 8.4 - 80}{\sqrt{2^2 + 8^2}}\:|}[/tex]
[tex]\mathsf{6 = |\:\dfrac{2k + 32 - 80}{\sqrt{4 + 64}}\:|}[/tex]
[tex]\mathsf{6 = \dfrac{2k - 48}{2\sqrt{17}}}[/tex]
[tex]\mathsf{12\sqrt{17} = 2k - 48}[/tex]
[tex]\mathsf{2k = 48 + 12\sqrt{17}}[/tex]
[tex]\boxed{\boxed{\mathsf{k = 24 + 6\sqrt{17}}}}[/tex]
[tex]\textsf{A(1;1) B(-1;-3) C(2;-7)}[/tex]
[tex]\textsf{reta suporte }\sf{\overline{BC}}:}[/tex]
[tex]\sf{m = \dfrac{\Delta_Y}{\Delta_X} = \dfrac{y_C - y_B}{x_C - x_B} = \dfrac{-7-(-3)}{2 - (-1)} = \dfrac{-7 + 3}{2 + 1} = -\dfrac{4}{3}}[/tex]
[tex]\mathsf{y - y_0 = m(x - x_0)}[/tex]
[tex]\mathsf{y - (-7) = -\dfrac{4}{3}(x - 2)}[/tex]
[tex]\mathsf{3y + 21 = -4x + 8}[/tex]
[tex]\sf{r:4x + 3y + 13 = 0}[/tex]
[tex]\textsf{reta perpendicular a }\sf{\overline{BC}}\textsf{ passando por A:}[/tex]
[tex]\mathsf{m_1.m_2 = -1}[/tex]
[tex]\mathsf{m_1.\left(-\dfrac{4}{3}\right) = -1}[/tex]
[tex]\sf{m = \dfrac{3}{4}}[/tex]
[tex]\mathsf{y - 1 = \dfrac{3}{4}(x - 1)}[/tex]
[tex]\mathsf{4y - 4 = 3x - 3}[/tex]
[tex]\mathsf{s:3x - 4y + 1 = 0}[/tex]
[tex]\textsf{ponto de concorr{\^e}ncia entre as retas r e s:}[/tex]
[tex]\begin{cases}\sf{4x + 3y = -13}\\\sf{3x - 4y = -1}\end{cases}[/tex]
[tex]\begin{cases}\sf{12x + 9y = -39}\\\sf{-12x + 16y = 4}\end{cases}[/tex]
[tex]\mathsf{25y = -35}[/tex]
[tex]\mathsf{y = -\dfrac{7}{5}}[/tex]
[tex]\mathsf{4x + 3\left(-\dfrac{7}{5}\right) = -13}[/tex]
[tex]\mathsf{20x - 21 = -65}[/tex]
[tex]\mathsf{20x = -44}[/tex]
[tex]\mathsf{x = -\dfrac{11}{5}}[/tex]
[tex]\mathsf{H\left(-\dfrac{11}{5};\:-\dfrac{7}{5}\right)}[/tex]
[tex]\textsf{calcular a dist{\^a}ncia entre A e H:}[/tex]
[tex]\sf{d_{AH} = \sqrt{(x_A - x_H)^2 + (y_A - y_H)^2}}}[/tex]
[tex]\sf{d_{AH} = \sqrt{(1 - (-11/5))^2 + (1 - (-7/5))^2}}}[/tex]
[tex]\sf{d_{AH} = \sqrt{(1 + (11/5))^2 + (1 + (7/5))^2}}}[/tex]
[tex]\sf{d_{AH} = \sqrt{(16/5)^2 + (12/5)^2}}}[/tex]
[tex]\sf{d_{AH} = \sqrt{(256/25) + (144/25)}}}[/tex]
[tex]\sf{d_{AH} = \sqrt{(400/25)}[/tex]
[tex]\sf{d_{AH} = \sqrt{16}[/tex]
[tex]\boxed{\boxed{\sf{d_{AH} = 4}}}\leftarrow\textsf{altura relativa ao v{\'e}rtice A}[/tex]
