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 5. \\(m (-7, 2)\\), \\(n (?, ?)\\), \\(p (28, -12)\\)

Explanation:

Step1: Recall the section formula

The section formula for a point \( N(x,y) \) dividing the line segment joining \( M(x_1,y_1) \) and \( P(x_2,y_2) \) in the ratio \( m:n \) is \( x=\frac{mx_2 + nx_1}{m + n} \) and \( y=\frac{my_2 + ny_1}{m + n} \). Here, \( m = 2 \), \( n = 5 \), \( x_1=-7 \), \( y_1 = 2 \), \( x_2=28 \), \( y_2=-12 \).

Step2: Calculate the x - coordinate of N

Substitute the values into the x - coordinate formula:
\( x=\frac{2\times28+5\times(-7)}{2 + 5}=\frac{56-35}{7}=\frac{21}{7}=3 \)

Step3: Calculate the y - coordinate of N

Substitute the values into the y - coordinate formula:
\( y=\frac{2\times(-12)+5\times2}{2 + 5}=\frac{-24 + 10}{7}=\frac{-14}{7}=-2 \)

Answer:

\((3,-2)\)