Question

a reta que passa pelos pontos A(m,2) e B(5,-m) tem coeficiente angular igual a -2/3. Qual o valor de m?

Answer 1

Dados dois pontos $A(x_{_{A}},\,y_{_{A}})$ e $B(x_{_{B}},\,y_{_{B}})\,,$ o coeficiente angular da reta que passa por $A$ e $B$ é dado por

$\dfrac{\Delta y}{\Delta x}=\dfrac{y_{_{B}}-y_{_{A}}}{x_{_{B}}-x_{_{A}}}~~~~~~(x_{_{B}}\ne x_{_{A}})$
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Para esta questão, temos $A(m,\,2)\,,B(5,\,-m)$ e o coeficiente angular é $-\frac{2}{3}.$ Então, devemos ter

$-\dfrac{2}{3}=\dfrac{y_{_{B}}-y_{_{A}}}{x_{_{B}}-x_{_{A}}}\\\\\\ -\dfrac{2}{3}=\dfrac{-m-2}{5-m}~~~~~~(m\ne 5)\\\\\\ -\dfrac{2}{3}=\dfrac{-(m+2)}{5-m}\\\\\\ \dfrac{2}{3}=\dfrac{m+2}{5-m}\\\\\\ 2\cdot (5-m)=3\cdot (m+2)\\\\ 10-2m=3m+6\\\\ -2m-3m=6-10\\\\ -5m=-4$

$m=\dfrac{-4}{-5}\\\\\\ \boxed{\begin{array}{c}m=\dfrac{4}{5} \end{array}}$