Question

Os pontos A (0, 0), B (3, 7)e C(5,-1)sao vértices de um triângulo. calcule comprimento da mediana BM

Answer 1

A mediana BM é o segmento que une o vértice B ao ponto médio do lado AC.

Vamos, então, começar determinando o ponto médio M do lado AC:

$M~=~\left(\frac{x_A+x_C}{2}~,~\frac{y_A+y_C}{2}\right)\\\\\\M~=~\left(\frac{0+5}{2}~,~\frac{0+(-1)}{2}\right)\\\\\\M~=~\left(\frac{5}{2}~,~\frac{-1}{2}\right)\\\\\\\boxed{M~=~(2,5~,\,-0,5)}$

Calculando, agora, o comprimento de BM pela distancia entre os pontos B e M, temos:

$Comprimento~\overline{BM}~=~Distancia_{\,B,M}\\\\\\Comprimento~\overline{BM}~=~\sqrt{(x_B-x_M)^2+(y_B-y_M)^2}\\\\\\Comprimento~\overline{BM}~=~\sqrt{(3-2,5)^2+(7-(-0,5))^2}\\\\\\Comprimento~\overline{BM}~=~\sqrt{(0,5)^2+(7,5)^2}\\\\\\Comprimento~\overline{BM}~=~\sqrt{0,25+56,25}\\\\\\Comprimento~\overline{BM}~=~\sqrt{56,50}\\\\\\Comprimento~\overline{BM}~=~\sqrt{\frac{5650}{100}}\\\\\\Comprimento~\overline{BM}~=~\frac{1}{10}~.~\sqrt{5650}$

$Comprimento~\overline{BM}~=~\frac{1}{10}~.~\sqrt{5~.~5~.~226}\\\\\\Comprimento~\overline{BM}~=~\frac{1}{10}~.~\sqrt{5^2~.~226}\\\\\\Comprimento~\overline{BM}~=~\frac{1}{10}~.~5~.~\sqrt{~226}\\\\\\Comprimento~\overline{BM}~=~\frac{1}{2}\sqrt{~226}\\\\\\\boxed{Comprimento~\overline{BM}~=~\frac{\sqrt{~226}}{2}~~unidades~de~comprimento}$