Question

Determine o simétrico de A em relação ao Q
a) A (-5,2) e Q (1/2,3)

b) A (3,8) e Q (-2,1)

Answer 1

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$\large\begin{array}{l}\textsf{De acordo com o enunciado da tarefa, deseja-se encontrar}\\\textsf{as coordenadas do ponto }\mathsf{B(x_{_B},\,y_{_B}),}\textsf{ de modo que} \end{array} \end{array}$


•   $\large\begin{array}{l} \textsf{Q \'e o ponto m\'edio do segmento }\overline{\mathsf{AB}}: \end{array}$

$\large\begin{array}{l} \mathsf{x_{_Q}=\dfrac{x_{_A}+x_{_B}}{2}}\\\\ \mathsf{2x_{_Q}=x_{_A}+x_{_B}}\\\\ \mathsf{x_{_B}=2x_{_Q}-x_{_A}\qquad\quad(i)}\end{array}$


$\large\begin{array}{l} \mathsf{y_{_Q}=\dfrac{y_{_A}+y_{_B}}{2}}\\\\ \mathsf{2y_{_Q}=y_{_A}+y_{_B}}\\\\ \mathsf{y_{_B}=2y_{_Q}-y_{_A}\qquad\quad(ii)} \end{array}$

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$\large\begin{array}{l} \textsf{a) Dados os pontos}\\\\ \mathsf{A(-5,\,2)~~e~~Q(\frac{1}{2},\,3),} \\\\\textsf{as coordenadas do ponto B, sim\'etrico de A}\\\textsf{em rela\c{c}\~ao a Q s\~ao:}\end{array}$


•   $\large\begin{array}{l} \mathsf{x_{_B}=2x_{_Q}-x_{_A}} \end{array}$

$\large\begin{array}{l} \mathsf{x_{_B}=2\cdot \dfrac{1}{2}-(-5)}\\\\ \mathsf{x_{_B}=1+5}\\\\ \mathsf{x_{_B}=6\qquad\quad\checkmark} \end{array}$


•   $\large\begin{array}{l} \mathsf{y_{_B}=2y_{_Q}-y_{_A}} \end{array}$

$\large\begin{array}{l} \mathsf{y_{_B}=2\cdot 3-2}\\\\ \mathsf{y_{_B}=6-2}\\\\ \mathsf{y_{_B}=4\qquad\quad\checkmark} \end{array}$


$\large\begin{array}{l} \textsf{O ponto procurado \'e o ponto }\mathsf{B(6,\,4).} \end{array}$

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$\large\begin{array}{l} \textsf{b) Dados os pontos}\\\\ \mathsf{A(3,\,8)~~e~~Q(-2,\,1),} \\\\\textsf{as coordenadas do ponto B, sim\'etrico de A}\\\textsf{em rela\c{c}\~ao a Q s\~ao:}\end{array}$


•   $\large\begin{array}{l} \mathsf{x_{_B}=2x_{_Q}-x_{_A}} \end{array}$

$\large\begin{array}{l} \mathsf{x_{_B}=2\cdot (-2)-3}\\\\ \mathsf{x_{_B}=-4-3}\\\\ \mathsf{x_{_B}=-7\qquad\quad\checkmark} \end{array}$


•   $\large\begin{array}{l} \mathsf{y_{_B}=2y_{_Q}-y_{_A}} \end{array}$

$\large\begin{array}{l} \mathsf{y_{_B}=2\cdot 1-8}\\\\ \mathsf{y_{_B}=2-8}\\\\ \mathsf{y_{_B}=-6\qquad\quad\checkmark} \end{array}$


$\large\begin{array}{l} \textsf{O ponto procurado \'e o ponto }\mathsf{B(-7,\,-6).} \end{array}$

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$\large\textsf{Bons estudos! :-)}$


Tags:  simétrico ponto médio simetria geometria analítica