Question

the point b lies on the segment \\(\overline{ac}\\). find the coordinates of b so that \\(ab\\) is \\(\frac{1}{7}\\) of \\(ac\\). a(-13, -20), c(1,1), b(?,?). coordinates of b : ( \\(\square\\) , \\(\square\\) )

Explanation:

Step1: Recall the section formula

If a point \( B(x,y) \) divides the line segment joining \( A(x_1,y_1) \) and \( C(x_2,y_2) \) in the ratio \( m:n \), then the coordinates of \( B \) are given by \( x=\frac{mx_2 + nx_1}{m + n} \) and \( y=\frac{my_2 + ny_1}{m + n} \). Here, \( AB=\frac{1}{7}AC \), so \( AB:BC = 1:(7 - 1)=1:6 \). So \( m = 1 \), \( n = 6 \), \( x_1=-13 \), \( y_1 = - 20 \), \( x_2 = 1 \), \( y_2=1 \).

Step2: Calculate the x - coordinate of B

Using the formula for \( x \)-coordinate: \( x=\frac{1\times1+6\times(-13)}{1 + 6}=\frac{1-78}{7}=\frac{-77}{7}=-11 \)

Step3: Calculate the y - coordinate of B

Using the formula for \( y \)-coordinate: \( y=\frac{1\times1+6\times(-20)}{1 + 6}=\frac{1-120}{7}=\frac{-119}{7}=-17 \)

Answer:

\((-11, -17)\)