To find the length of the median from vertex [tex]\( B \)[/tex] to side [tex]\( AC \)[/tex] in triangle [tex]\( ABC \)[/tex] with vertices [tex]\( A(-6, 7) \)[/tex], [tex]\( B(4, -1) \)[/tex], and [tex]\( C(-2, -9) \)[/tex], we need to follow these steps:
1. Find the midpoint [tex]\( M \)[/tex] of side [tex]\( AC \)[/tex].
The formula for the midpoint [tex]\( M \)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
For points [tex]\( A(-6, 7) \)[/tex] and [tex]\( C(-2, -9) \)[/tex], we apply the midpoint formula:
[tex]\[
M = \left( \frac{-6 + (-2)}{2}, \frac{7 + (-9)}{2} \right) = \left( \frac{-8}{2}, \frac{-2}{2} \right) = (-4, -1)
\][/tex]
2. Calculate the length of the median [tex]\( BM \)[/tex].
The length of a line segment between two points [tex]\( B(x_1, y_1) \)[/tex] and [tex]\( M(x_2, y_2) \)[/tex] is found using the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here, [tex]\( B(4, -1) \)[/tex] and [tex]\( M(-4, -1) \)[/tex], so:
[tex]\[
BM = \sqrt{(-4 - 4)^2 + (-1 - (-1))^2} = \sqrt{(-8)^2 + (0)^2} = \sqrt{64} = 8
\][/tex]
Thus, the length of the median from vertex [tex]\( B \)[/tex] to side [tex]\( AC \)[/tex] is [tex]\(\boxed{8}\)[/tex].
