Question

3)(PUC)O ponto B = (3. b) é equidistante dos pontos A = (3.0) e C=(0, 2), Logo, o ponto B é:

Answer 1

✅ Após resolver os cálculos, concluímos que o ponto "B" procurado é:

                  $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf B\bigg(3,\,\frac{13}{4}\bigg)\:\:\:}}\end{gathered} }$

Sejam os pontos:

                           $\Large\begin{cases} A(3, 0)\\B(3, b)\\C(0, 2)\end{cases}$

Se o ponto "B" é equidistante dos pontos "A" e "C", significa dizer que a distância entre os pontos A e B é igual à distância entre os pontos B e C. Deste modo, temos:

                                       $\Large\displaystyle\begin{gathered} d(AB) = d(AC)\end{gathered}$

     $\large\displaystyle\text{ \begin{gathered} \sqrt{(x_{B} - x_{A})^{2} + (y_{B} - y_{A})^{2}} = \sqrt{(x_{C} - x_{A})^{2} + (y_{C} - y_{A})^{2}}\end{gathered} }$

$\large\displaystyle\text{ \begin{gathered} (\sqrt{(x_{B} - x_{A})^{2} + (y_{B} - y_{A})^{2}})^{2} = (\sqrt{(x_{C} - x_{A})^{2} + (y_{C} - y_{A})^{2}})^{2}\end{gathered} }$

$\Large\displaystyle\begin{gathered} (x_{B} - x_{A})^{2} + (y_{B} - y_{A})^{2} = (x_{C} - x_{A})^{2} + (y_{C} - y_{A})^{2}\end{gathered}$

             $\Large\displaystyle\begin{gathered} (3 - 3)^{2} + (b - 0)^{2} = (0 - 3)^{2} + (2 - b)^{2}\end{gathered}$

                                       $\Large\displaystyle\begin{gathered} 0^{2} + b^{2} = (-3)^{2} + 4 - 4b + b^{2}\end{gathered}$

                                                 $\Large\displaystyle\begin{gathered} b^{2} = 9 + 4 - 4b + b^{2}\end{gathered}$

          $\Large\displaystyle\begin{gathered} b^{2} - 9 - 4 + 4b - b^{2} = 0\end{gathered}$

                           $\Large\displaystyle\begin{gathered} -9 - 4 + 4b = 0\end{gathered}$

                                                $\Large\displaystyle\begin{gathered} 4b = 9 + 4\end{gathered}$

                                                $\Large\displaystyle\begin{gathered} 4b = 13\end{gathered}$

                                                  $\Large\displaystyle\begin{gathered} b = \frac{13}{4}\end{gathered}$

Portanto, o valor da ordenada do ponto "B" é:

                                                  $\Large\displaystyle\begin{gathered} b = \frac{13}{4}\end{gathered}$

✅ Portanto, o ponto "B" é:

                                             $\Large\displaystyle\begin{gathered} B\bigg(3,\,\frac{13}{4}\bigg)\end{gathered}$

$\LARGE\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered} }$

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$\Large\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Observe \:o\:Gr\acute{a}fico!!\:\:\:}}}\end{gathered} }$