To find the coordinates of point D, which divides the line segment BC in the ratio BD:CD = 7:1, we can use the section formula.
Let the coordinates of point B be \( B(-11, 6) \) and the coordinates of point C be \( C(5, -10) \). The formula for finding the coordinates of a point that divides a line segment in the ratio \( m:n \) is given by:
\[ D\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
In this case, \( m = 7 \) and \( n = 1 \), \( (x_1, y_1) = (-11, 6) \), and \( (x_2, y_2) = (5, -10) \).
Now we substitute these values into the formula:
- Calculate the x-coordinate of D:
\[ x_D = \frac{7 \cdot 5 + 1 \cdot (-11)}{7 + 1} = \frac{35 - 11}{8} = \frac{24}{8} = 3 \]
- Calculate the y-coordinate of D:
\[ y_D = \frac{7 \cdot (-10) + 1 \cdot 6}{7 + 1} = \frac{-70 + 6}{8} = \frac{-64}{8} = -8 \]
Therefore, the coordinates of point D are:
\[ D(3, -8) \]