Question

Dada a reta r:3x+4y-15=0, calcule a distância ao ponto P(0,5)

Answer 1

Basta utilizarmos a equação da distancia de um ponto a uma reta:

$Distancia_{\,P\,,\,r}~=~\frac{\left|a\,.\,x_{_P}~+~b\,.\,y_{_P}~+~c\right|}{\sqrt{a^2+b^2}}\\\\\\Na~reta~"r",~os~coeficicentes~"a",~"b"~e~"c"~sao~,respectivamente,~3,~4~e~-15\\\\No~ponto~P,~as~coordenadas~~ x_{_P}~e~y_{_P}~sao,~respectivamente,~0~e~5\\\\\\Substituindo\\\\\\Distancia_{\,P\,,\,r}~=~\frac{\left|3\,.\,0~+~4\,.\,5~+~(-15)\right|}{\sqrt{3^2+4^2}}\\\\\\Distancia_{\,P\,,\,r}~=~\frac{\left|0~+~20~-~15\right|}{\sqrt{9+16}}$

$Distancia_{\,P\,,\,r}~=~\frac{\left|5\right|}{\sqrt{25}}\\\\\\Distancia_{\,P\,,\,r}~=~\frac{5}{5}\\\\\\\boxed{Distancia_{\,P\,,\,r}~=~1~unidade}$