To find the length of the median from vertex [tex]\( B \)[/tex] in triangle [tex]\( ABC \)[/tex] with vertices [tex]\( A(-6, 7) \)[/tex], [tex]\( B(4, -1) \)[/tex], and [tex]\( C(-2, -9) \)[/tex], follow these steps:
1. Identify the coordinates of the midpoint of segment [tex]\( AC \)[/tex]:
[tex]\[
\text{Midpoint of } AC = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\][/tex]
where [tex]\( A(x_1, y_1) = A(-6, 7) \)[/tex] and [tex]\( C(x_2, y_2) = C(-2, -9) \)[/tex].
Calculate the midpoint:
[tex]\[
\text{Midpoint of } AC = \left(\frac{-6 + (-2)}{2}, \frac{7 + (-9)}{2}\right) = \left(\frac{-8}{2}, \frac{-2}{2}\right) = (-4, -1)
\][/tex]
2. Calculate the length of the median from [tex]\( B \)[/tex] to the midpoint of [tex]\( AC \)[/tex]:
Use the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
where [tex]\( B(x_1, y_1) = B(4, -1) \)[/tex] and the midpoint [tex]\( M(x_2, y_2) = (-4, -1) \)[/tex].
Calculate the distance:
[tex]\[
\text{Distance} = \sqrt{(4 - (-4))^2 + (-1 - (-1))^2}
\][/tex]
Simplify inside the square root:
[tex]\[
\sqrt{(4 + 4)^2 + (0)^2} = \sqrt{8^2 + 0} = \sqrt{64} = 8
\][/tex]
Therefore, the length of the median from [tex]\( B \)[/tex] to the midpoint of [tex]\( AC \)[/tex] is:
[tex]\[ \boxed{8} \][/tex]
