Question

the point n lies on the segment (overline{mp}). find the coordinates of n so that the ratio of mn to np is 1 to 3.
p (19,10)
n (?, ?)
m (-1, -2)

Explanation:

Step1: Recall 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 \( x=\frac{mx_2 + nx_1}{m + n} \), \( y=\frac{my_2+ny_1}{m + n} \). Here, \( m = 1 \), \( n = 3 \), \( x_1=-1,y_1 = - 2,x_2 = 19,y_2=10 \).

Step2: Calculate x-coordinate of N

Substitute values into x - formula: \( x=\frac{1\times19+3\times(-1)}{1 + 3}=\frac{19-3}{4}=\frac{16}{4}=4 \)

Step3: Calculate y-coordinate of N

Substitute values into y - formula: \( y=\frac{1\times10+3\times(-2)}{1 + 3}=\frac{10 - 6}{4}=\frac{4}{4}=1 \)

Answer:

The coordinates of \( N \) are \( (4,1) \)