{"id":73,"date":"2023-09-16T13:01:17","date_gmt":"2023-09-16T13:01:17","guid":{"rendered":"https:\/\/mathority.org\/pt\/formula-para-o-ponto-medio-de-um-vetor-de-segmento\/"},"modified":"2023-09-16T13:01:17","modified_gmt":"2023-09-16T13:01:17","slug":"formula-para-o-ponto-medio-de-um-vetor-de-segmento","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/formula-para-o-ponto-medio-de-um-vetor-de-segmento\/","title":{"rendered":"F\u00f3rmula para o meio de um segmento"},"content":{"rendered":"
Esta p\u00e1gina explica o significado do ponto m\u00e9dio de um segmento. Al\u00e9m disso, voc\u00ea descobrir\u00e1 como encontrar o meio de um segmento usando sua f\u00f3rmula. Voc\u00ea ainda ver\u00e1 exemplos, exerc\u00edcios e problemas resolvidos de pontos m\u00e9dios de segmentos. <\/p>\n
\n
<\/div>\n<\/div>\n
<\/span> Qual \u00e9 o ponto m\u00e9dio de um segmento? <\/span><\/h2>\n
\n
<\/div>\n<\/div>\n
Em matem\u00e1tica, o ponto m\u00e9dio de um segmento<\/strong> \u00e9 o ponto localizado \u00e0 mesma dist\u00e2ncia das extremidades de um segmento. O meio, portanto, divide o segmento em duas partes iguais. <\/p>\n
\n<\/figure>\n<\/div>\n
Al\u00e9m disso, o ponto m\u00e9dio est\u00e1 bem no centro do segmento, portanto pertence \u00e0 bissetriz do segmento.<\/p>\n
Por outro lado, o ponto m\u00e9dio de um segmento \u00e9 tamb\u00e9m um ponto equidistante de dois elementos geom\u00e9tricos: as duas extremidades do segmento. <\/p>\n
<\/span> Como calcular o ponto m\u00e9dio de um segmento? <\/span><\/h2>\n
\n
<\/div>\n<\/div>\n
\n
Dadas as coordenadas cartesianas dos pontos extremos de um segmento:<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
As coordenadas do meio do referido segmento correspondem \u00e0 meia soma das coordenadas dos pontos extremos:<\/p>\n<\/p>\n
<\/p>\n<\/p>\n<\/div>\n
Esta \u00e9 a f\u00f3rmula para o meio de um segmento no plano cartesiano (em R2). Mas obviamente a f\u00f3rmula tamb\u00e9m \u00e9 aplic\u00e1vel ao espa\u00e7o cartesiano (em R3), bastando somar a meia soma da coordenada Z: <\/p>\n
\n<\/figure>\n<\/div>\n
Vejamos um exemplo de como calcular as coordenadas do ponto m\u00e9dio de um segmento:<\/p>\n
\n
Determine o ponto m\u00e9dio do segmento formado pelos seguintes pontos:<\/li>\n<\/ul>\n
<\/p>\n<\/p>\n
Para encontrar o meio do segmento, basta aplicar sua f\u00f3rmula: <\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/span> Exerc\u00edcios resolvidos no meio de um segmento <\/span><\/h2>\n
\n
<\/div>\n<\/div>\n
Exerc\u00edcio 1<\/h3>\n
Qual \u00e9 o ponto m\u00e9dio do segmento cujas extremidades s\u00e3o os dois pontos seguintes? <\/p>\n<\/p>\n
<\/p>\n<\/p>\n
\n
Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n
Para encontrar o meio do segmento voc\u00ea deve aplicar diretamente a f\u00f3rmula:<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/div>\n
Exerc\u00edcio 2<\/h3>\n
Encontre as coordenadas do ponto final do segmento que come\u00e7a no ponto A e cujo ponto m\u00e9dio \u00e9 M. <\/p>\n<\/p>\n
<\/p>\n<\/p>\n
\n
Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n
Neste caso conhecemos as coordenadas do ponto inicial e do meio do segmento. Portanto, substitu\u00edmos as coordenadas que conhecemos na f\u00f3rmula do ponto m\u00e9dio de um segmento: <\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
E agora resolvemos as coordenadas do ponto final do segmento da equa\u00e7\u00e3o anterior: <\/p>\n
\n
\n
Coordenadas X<\/span> <\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n<\/div>\n
\n
Coordenadas Y<\/span> <\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n<\/div>\n<\/div>\n
As coordenadas da extremidade final do segmento s\u00e3o, portanto: <\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/div>\n
Exerc\u00edcio 3<\/h3>\n
Dado o seguinte paralelogramo: <\/p>\n
\n<\/figure>\n<\/div>\n
Sabemos que M \u00e9 o centro do paralelogramo e as coordenadas dos pontos A, B e C s\u00e3o:<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
A partir dessas informa\u00e7\u00f5es e utilizando a f\u00f3rmula do ponto m\u00e9dio, calcule as coordenadas do ponto D. <\/p>\n
\n
Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n
Para encontrar as coordenadas do ponto D usando a f\u00f3rmula do meio de um segmento, voc\u00ea deve primeiro calcular as coordenadas do ponto M e depois as do ponto D.<\/p>\n
O ponto M \u00e9 o ponto m\u00e9dio do segmento BC, suas coordenadas s\u00e3o portanto: <\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
E uma vez conhecido o ponto M, podemos encontrar o ponto D. O ponto M tamb\u00e9m \u00e9 o meio do segmento AD, ent\u00e3o:<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
\n
\n
Coordenada X do ponto D<\/span> <\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n<\/div>\n
\n
Coordenada Y do ponto D<\/span> <\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/p>\n<\/p>\n<\/div>\n<\/div>\n
As coordenadas do ponto D s\u00e3o, portanto: <\/p>\n<\/p>\n
<\/p>\n<\/p>\n
<\/div>\n
Exerc\u00edcio 4<\/h3>\n
Calcule a equa\u00e7\u00e3o cont\u00ednua da reta perpendicular ao segmento PQ em seu ponto m\u00e9dio. Seja os pontos<\/p>\n<\/p>\n
<\/p>\n
E <\/p>\n
<\/p>\n<\/p>\n
\n
Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n
Para determinar a equa\u00e7\u00e3o de uma reta, precisamos de seu vetor diretor e de um ponto que faz parte da reta.<\/p>\n
Neste caso, o vetor de dire\u00e7\u00e3o da reta ser\u00e1 perpendicular ao vetor<\/p>\n<\/p>\n
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