Question

the point b lies on the segment (overline{ac}). find the coordinates of b so that the ratio of ab to bc is 1 to 6.

(image: coordinate plane with point a(-13, -20), point c(1,1), and segment ac with point b between them. input box for coordinates of b: ( , ))

Explanation:

Step1: Recall the section formula

The section formula for a point \( B(x,y) \) dividing the line segment joining \( A(x_1,y_1) \) and \( C(x_2,y_2) \) in the ratio \( m:n \) is given by \( x=\frac{mx_2 + nx_1}{m + n} \) and \( y=\frac{my_2 + ny_1}{m + n} \). Here, \( m = 1 \), \( n = 6 \), \( A(-13,-20) \) and \( C(1,1) \).

Step2: Calculate the x - coordinate of B

Substitute \( m = 1 \), \( n = 6 \), \( x_1=-13 \), \( x_2 = 1 \) into the formula for \( x \):
\( x=\frac{1\times1+6\times(-13)}{1 + 6}=\frac{1-78}{7}=\frac{-77}{7}=-11 \)

Step3: Calculate the y - coordinate of B

Substitute \( m = 1 \), \( n = 6 \), \( y_1=-20 \), \( y_2 = 1 \) into the formula for \( y \):
\( y=\frac{1\times1+6\times(-20)}{1 + 6}=\frac{1-120}{7}=\frac{-119}{7}=-17 \)

Answer:

\((-11,-17)\)