Question

point p ( , ) is \\(\frac{4}{5}\\) the distance from a to b. write your answer in decimal form.
chart: coordinate grid with point a (-4, 2) and point b (1, 8) connected by a line segment

Explanation:

Step1: Recall the section formula

The coordinates of a point \( P(x,y) \) that divides the line segment joining \( A(x_1,y_1) \) and \( B(x_2,y_2) \) in the ratio \( m:n \) are given by \( x=\frac{mx_2 + nx_1}{m + n} \) and \( y=\frac{my_2 + ny_1}{m + n} \). Here, since \( P \) is \( \frac{4}{5} \) the distance from \( A \) to \( B \), the ratio \( m:n = 4:1 \) (because the remaining distance from \( P \) to \( B \) is \( 1-\frac{4}{5}=\frac{1}{5} \), so \( m = 4 \), \( n = 1 \)). \( A(-4,2) \) so \( x_1=-4,y_1 = 2 \), \( B(1,8) \) so \( x_2 = 1,y_2=8 \).

Step2: Calculate the x - coordinate of \( P \)

Using the formula for \( x \):
\( x=\frac{4\times1+1\times(-4)}{4 + 1}=\frac{4-4}{5}=\frac{0}{5}=0 \)

Step3: Calculate the y - coordinate of \( P \)

Using the formula for \( y \):
\( y=\frac{4\times8+1\times2}{4 + 1}=\frac{32 + 2}{5}=\frac{34}{5}=6.8 \)

Answer:

\( (0, 6.8) \)