Question

the coordinates of the endpoints of \\(overline{ab}\\) are \\(a(-8, -8)\\) and \\(b(4, -2)\\). point \\(c\\) is on \\(overline{ab}\\) and divides it such that \\(ac:bc\\) is \\(5:1\\). what are the coordinates of \\(c\\)? write your answers as integers or decimals. (\\(square\\), \\(square\\)) submit

Explanation:

Step1: Recall the section formula

The section formula for a point \( C(x,y) \) dividing the line segment joining \( A(x_1,y_1) \) and \( B(x_2,y_2) \) in the ratio \( m:n \) is given by:
\[
x=\frac{mx_2 + nx_1}{m + n}, \quad y=\frac{my_2 + ny_1}{m + n}
\]
Here, \( A(-8,-8) \), \( B(4,-2) \), \( m = 5 \), \( n = 1 \).

Step2: Calculate the x - coordinate of C

Substitute \( x_1=-8 \), \( x_2 = 4 \), \( m = 5 \), \( n = 1 \) into the formula for \( x \):
\[
x=\frac{5\times4+1\times(-8)}{5 + 1}=\frac{20-8}{6}=\frac{12}{6}=2
\]

Step3: Calculate the y - coordinate of C

Substitute \( y_1=-8 \), \( y_2=-2 \), \( m = 5 \), \( n = 1 \) into the formula for \( y \):
\[
y=\frac{5\times(-2)+1\times(-8)}{5 + 1}=\frac{-10-8}{6}=\frac{-18}{6}=-3
\]

Answer:

\((2, - 3)\)