Question

the coordinates of the endpoints of \\(\overline{bc}\\) are \\(b(-11, 6)\\) and \\(c(5, -10)\\). point \\(d\\) is on \\(\overline{bc}\\) and divides it such that \\(bd:cd\\) is \\(7:1\\). what are the coordinates of \\(d\\)? write your answers as integers or decimals. (\\(\square\\), \\(\square\\)) submit

Explanation:

To find the coordinates of point \( D \) that divides the line segment \( \overline{BC} \) in the ratio \( BD:CD = 7:1 \), we can use the section formula. The section formula for a point \( (x, y) \) dividing the line segment joining \( (x_1, y_1) \) and \( (x_2, y_2) \) 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, \( B(-11, 6) \) is \( (x_1, y_1) \), \( C(5, -10) \) is \( (x_2, y_2) \), and the ratio \( m:n = 7:1 \).

Step 1: Calculate the x-coordinate of \( D \)

Substitute \( x_1 = -11 \), \( x_2 = 5 \), \( m = 7 \), and \( n = 1 \) into the formula for \( x \):

\[
x = \frac{7 \times 5 + 1 \times (-11)}{7 + 1}
\]

\[
x = \frac{35 - 11}{8}
\]

\[
x = \frac{24}{8}
\]

\[
x = 3
\]

Step 2: Calculate the y-coordinate of \( D \)

Substitute \( y_1 = 6 \), \( y_2 = -10 \), \( m = 7 \), and \( n = 1 \) into the formula for \( y \):

\[
y = \frac{7 \times (-10) + 1 \times 6}{7 + 1}
\]

\[
y = \frac{-70 + 6}{8}
\]

\[
y = \frac{-64}{8}
\]

\[
y = -8
\]

Answer:

The coordinates of \( D \) are \( (3, -8) \).