Question

the coordinates of the endpoints of \\( \overline{mn} \\) are \\( m(-8, 12) \\) and \\( n(1, -6) \\). point \\( o \\) is on \\( \overline{mn} \\) and divides it such that \\( mo:no \\) is \\( 4:5 \\). what are the coordinates of \\( o \\)? write your answers as integers or decimals. \\( (\quad, \quad) \\)

Explanation:

Step1: Recall the section formula

When a point \( O(x,y) \) divides a line segment joining \( M(x_1,y_1) \) and \( N(x_2,y_2) \) in the ratio \( m:n \), the coordinates of \( O \) are given by \( x=\frac{mx_2 + nx_1}{m + n} \) and \( y=\frac{my_2+ny_1}{m + n} \). Here, \( MO:NO = 4:5 \), so \( m = 4 \), \( n = 5 \), \( x_1=-8,y_1 = 12,x_2 = 1,y_2=-6 \).

Step2: Calculate the x - coordinate of O

Substitute the values into the formula for \( x \):
\( x=\frac{4\times1+5\times(-8)}{4 + 5}=\frac{4-40}{9}=\frac{-36}{9}=-4 \)

Step3: Calculate the y - coordinate of O

Substitute the values into the formula for \( y \):
\( y=\frac{4\times(-6)+5\times12}{4 + 5}=\frac{-24 + 60}{9}=\frac{36}{9}=4 \)

Answer:

\((-4,4)\)