To calculate the median of the triangle passing through point A, follow these steps:
1. Identify the Midpoint of Line Segment BC: The median from a vertex (A in this case) of a triangle goes to the midpoint of the opposite side (BC here). To find the midpoint [tex]\( M \)[/tex] of segment BC, we use the midpoint formula:
[tex]\[
M = \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2} \right)
\][/tex]
For points B(4, -1) and C(6, 2):
[tex]\[
M_x = \frac{4 + 6}{2} = 5
\][/tex]
[tex]\[
M_y = \frac{-1 + 2}{2} = 0.5
\][/tex]
So, the coordinates of the midpoint [tex]\( M \)[/tex] are [tex]\( (5.0, 0.5) \)[/tex].
2. Calculate the Length of the Median from A to M: The median length is the distance between point A(3, 2) and the midpoint [tex]\( M(5.0, 0.5) \)[/tex]. The distance formula is:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting the coordinates:
[tex]\[
\text{Distance} = \sqrt{(5.0 - 3)^2 + (0.5 - 2)^2}
\][/tex]
[tex]\[
= \sqrt{(2.0)^2 + (-1.5)^2}
\][/tex]
[tex]\[
= \sqrt{4 + 2.25}
\][/tex]
[tex]\[
= \sqrt{6.25}
\][/tex]
[tex]\[
= 2.5
\][/tex]
3. Summary:
- The midpoint of segment BC is [tex]\( (5.0, 0.5) \)[/tex].
- The length of the median from point A to the midpoint [tex]\( M \)[/tex] is 2.5 units.
Therefore, the median from point A(3, 2) of the triangle passes through the midpoint (5.0, 0.5) and the length of this median is 2.5 units.
