Question

persevere point c (6,9) is located on the segment between point a (4,8) and point b. point c is \\(\frac{1}{4}\\) of the distance from a to b. what are the coordinates of point b? ( , ) need help?

Explanation:

Step1: Recall the section formula

If a point \( C(x,y) \) divides the line segment joining \( A(x_1,y_1) \) and \( B(x_2,y_2) \) in the ratio \( m:n \), then \( x=\frac{mx_2+nx_1}{m + n} \) and \( y=\frac{my_2+ny_1}{m + n} \). Here, \( C \) is \( \frac{1}{4} \) of the distance from \( A \) to \( B \), so the ratio \( m:n=1:3 \) (since \( AC:CB = 1:3 \)). Given \( A(4,8) \), \( C(6,9) \), let \( B=(x_2,y_2) \).

Step2: Solve for \( x_2 \)

Using the \( x \)-coordinate formula: \( 6=\frac{1\times x_2+3\times4}{1 + 3} \)
Simplify: \( 6=\frac{x_2 + 12}{4} \)
Multiply both sides by 4: \( 24=x_2+12 \)
Subtract 12: \( x_2 = 12 \)

Step3: Solve for \( y_2 \)

Using the \( y \)-coordinate formula: \( 9=\frac{1\times y_2+3\times8}{1 + 3} \)
Simplify: \( 9=\frac{y_2 + 24}{4} \)
Multiply both sides by 4: \( 36=y_2+24 \)
Subtract 24: \( y_2 = 12 \)

Answer:

\((12, 12)\)