What are the coordinates of the point that is 1/6 of the way from A(14,-1) to B(-4,23)? A. (8,6)B. (5,11) C. (-1,19)D. (11,3)​

Accepted Solution

Answer:D. (11,3)​Step-by-step explanation:First of all, let's plot each point as indicated in figure below. To find the point [tex]P(x,y)[/tex] that divide a segment with endpoints [tex]P_{1}(x_{1},y_{1}) \ and \ P_{2}(x_{2},y_{2})[/tex] and a ratio:[tex]r=\frac{\left | \overline{P_{1}P}\right |}{\left | \overline{PP_{2}} \right |}[/tex]We must use the formula:[tex](x,y)=\left(\frac{x_{1}+rx_{2}}{1+r},\frac{y_{1}+ry_{2}}{1+r}\right)[/tex]To find the ratio, we know that:[tex]\left | \overline{P_{1}P}\right |=\frac{1}{6}[/tex]because [tex]P[/tex] is 1/6 of the way from [tex]A(14,-1)[/tex] to [tex]B(-4,23)[/tex]So the other part of the segment is:[tex]\left | \overline{PP_{2}} \right |=\frac{5}{6}[/tex]Therefore, the ratio can be found as:[tex]r=\frac{\left | \overline{P_{1}P}\right |}{\left | \overline{PP_{2}} \right |}=\frac{\frac{1}{6}}{\frac{5}{6}}=\frac{1}{5}[/tex]From here, we can calculate the point we are looking for:[tex](x,y)=\left(\frac{14+(1/5)(-4)}{1+1/5},\frac{-1+(1/5)(23)}{1+1/5}\right) \\ \\ (x,y)=(11,3) \\ \\ \\ Finally: \\ \\ \boxed{P(x,y)=P(11,3)}[/tex]