Question

the point n lies on the segment mp. find the coordinates of n so that mn is $\frac{4}{9}$ of mp. p (21, 13) n (?,?) m (-6, -5) coordinates of n : (, )

Explanation:

Step1: Use section - formula for x - coordinate

Let \(M(x_1,y_1)=(-6,-5)\) and \(P(x_2,y_2)=(21,13)\). The formula for the x - coordinate of a point \(N\) that divides the line - segment \(MP\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\). Here, \(m = 4\) and \(n=9 - 4=5\).
\[x=\frac{4\times21 + 5\times(-6)}{4 + 5}=\frac{84-30}{9}=\frac{54}{9}=6\]

Step2: Use section - formula for y - coordinate

The formula for the y - coordinate of a point \(N\) that divides the line - segment \(MP\) in the ratio \(m:n\) is \(y=\frac{my_2+ny_1}{m + n}\).
\[y=\frac{4\times13+5\times(-5)}{4 + 5}=\frac{52 - 25}{9}=\frac{27}{9}=3\]

Answer:

\((6,3)\)