Question

6. Considere os pontos A(2 , 2) e B( -3 , -5) , calcule:
a) A distância entre esses dois pontos
b) O ponto médio do segmento que contém essas extremidades
c) A equação , na forma geral e reduzida, da reta que passa pelos pontos A e B.

Answer 1

a)

$\mathsf{{(x_{B}-x_{A})}^{2}={(-3-2)}^{2}={(-5)}^{2}=25}\\\mathsf{{(y_{B}-y_{A})}^{2}={(-5-2\\)}^{2}={(-7)}^{2}=49}$

$\mathsf{d_{A,B}=\sqrt{{(x_{B}-x_{A})}^{2}+{(y_{B}-y_{A})}^{2}}}\\\mathsf{d_{A,B}=\sqrt{25+49}=\sqrt{74}}$

b)

$\mathsf{x_{M}=\dfrac{x_{A}+x_{B}}{2}=\dfrac{2-3}{2}=-\dfrac{1}{2}}\\\mathsf{y_{M}=\dfrac{y_{A}+y_{B}}{2}=\dfrac{2-5}{2}=-\dfrac{3}{2}}\\\huge\boxed{\boxed{\mathsf{M(-\dfrac{1}{2},-\dfrac{3}{2})}}}$

c)

$\mathsf{m=\dfrac{y_{B}-y_{A}}{x_{B}-x_{A}}=\dfrac{-5-2}{-3-2}=\dfrac{7}{5}}$

$\mathsf{y=y_{0}+m(x-x_{0})}\\\mathsf{y=2+\dfrac{7}{5}(x-2)}\\\mathsf{y=2+\dfrac{7x}{5}-\dfrac{14}{5}}\\\mathsf{y=\dfrac{7x-4}{5}~eq\,reduzida}\\\mathsf{5y=7x-4}\\\mathsf{7x-5y-4=0\to~eq\,geral}$