Certainly! To determine the length of the median from vertex [tex]\( A(1,4) \)[/tex] to the midpoint of the side [tex]\( BC \)[/tex] in the triangle with vertices [tex]\( A(1,4) \)[/tex], [tex]\( B(-3,-2) \)[/tex], and [tex]\( C(3,0) \)[/tex], follow these steps:
1. Find the Midpoint of Side [tex]\( BC \)[/tex]:
The formula to find 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:
[tex]\[
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\][/tex]
For points [tex]\( B(-3, -2) \)[/tex] and [tex]\( C(3, 0) \)[/tex]:
[tex]\[
\text{Midpoint } M = \left(\frac{-3 + 3}{2}, \frac{-2 + 0}{2}\right) = (0.0, -1.0)
\][/tex]
2. Calculate the Length of the Median:
The median from vertex [tex]\( A \)[/tex] to the midpoint of [tex]\( BC \)[/tex] is the distance between the points [tex]\( A(1, 4) \)[/tex] and [tex]\( M(0.0, -1.0) \)[/tex].
The distance formula between two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Using the coordinates [tex]\( A(1, 4) \)[/tex] and [tex]\( M(0.0, -1.0) \)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(0.0 - 1)^2 + (-1.0 - 4)^2}
\][/tex]
[tex]\[
= \sqrt{(-1)^2 + (-5)^2}
\][/tex]
[tex]\[
= \sqrt{1 + 25}
\][/tex]
[tex]\[
= \sqrt{26}
\][/tex]
[tex]\[
= 5.0990195135927845
\][/tex]
Thus, the midpoint of side [tex]\( BC \)[/tex] is at [tex]\( (0.0, -1.0) \)[/tex], and the length of the median from vertex [tex]\( A \)[/tex] to this midpoint is approximately [tex]\( 5.099 \)[/tex].
