Question

the point n lies on the segment $overline{mp}$. find the coordinates of n so that the ratio of mn to np is 2 to 3. p (2, 7) n (?,?) coordinates of n : try one last time recheck

Explanation:

1. Assume the coordinates of \(M\) are \((x_1,y_1)\) and of \(P\) are \((x_2,y_2)=(2,7)\), and the ratio \(m:n = 2:3\). The section - formula for a point \(N(x,y)\) that divides the line - segment joining \((x_1,y_1)\) and \((x_2,y_2)\) in the ratio \(m:n\) is given by:

• \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Since the starting - point \(M\) is not given, assume \(M=(x_1,y_1)=(0,0)\) (if no other information about \(M\) is provided, and for the general formula application). Here \(m = 2\) and \(n = 3\), \(x_2=2\), \(y_2 = 7\).

1. Calculate the \(x\) - coordinate of \(N\):

• Using the formula \(x=\frac{mx_2+nx_1}{m + n}\), substitute \(m = 2\), \(n = 3\), \(x_1 = 0\), and \(x_2=2\).
• \(x=\frac{2\times2+3\times0}{2 + 3}=\frac{4+0}{5}=\frac{4}{5}=0.8\).

1. Calculate the \(y\) - coordinate of \(N\):

• Using the formula \(y=\frac{my_2+ny_1}{m + n}\), substitute \(m = 2\), \(n = 3\), \(y_1 = 0\), and \(y_2=7\).
• \(y=\frac{2\times7+3\times0}{2 + 3}=\frac{14+0}{5}=\frac{14}{5}=2.8\).

Answer:

\((0.8,2.8)\)