To determine the equation of the median from vertex [tex]\( J \)[/tex] for the triangle with vertices [tex]\( J(2, 5) \)[/tex], [tex]\( K(4, -1) \)[/tex], and [tex]\( L(-2, -5) \)[/tex], we will follow these steps:
1. Find the midpoint of the line segment [tex]\( KL \)[/tex].
2. Calculate the slope of the line that joins vertex [tex]\( J \)[/tex] and the midpoint.
3. Use the point-slope form of the equation of a line to find the equation in slope-intercept form.
### Step 1: Find the midpoint of [tex]\( KL \)[/tex]
The coordinates of the midpoint [tex]\( M \)[/tex] of a line segment between two points [tex]\( K(x_1, y_1) \)[/tex] and [tex]\( L(x_2, y_2) \)[/tex] are given by:
[tex]\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
For points [tex]\( K(4, -1) \)[/tex] and [tex]\( L(-2, -5) \)[/tex]:
[tex]\[
M_x = \frac{4 + (-2)}{2} = \frac{2}{2} = 1
\][/tex]
[tex]\[
M_y = \frac{-1 + (-5)}{2} = \frac{-6}{2} = -3
\][/tex]
So, the midpoint [tex]\( M \)[/tex] is [tex]\((1, -3)\)[/tex].
### Step 2: Calculate the slope of the median
The slope [tex]\( m \)[/tex] of the line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Using the points [tex]\( J(2, 5) \)[/tex] and [tex]\( M(1, -3) \)[/tex]:
[tex]\[
m = \frac{-3 - 5}{1 - 2} = \frac{-8}{-1} = 8
\][/tex]
The slope of the median line is [tex]\( 8 \)[/tex].
### Step 3: Use the point-slope form to find the equation of the line
The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Using the slope [tex]\( 8 \)[/tex] and the point [tex]\( J(2, 5) \)[/tex]:
[tex]\[
y - 5 = 8(x - 2)
\][/tex]
Rearrange this to the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y - 5 = 8x - 16
\][/tex]
[tex]\[
y = 8x - 16 + 5
\][/tex]
[tex]\[
y = 8x - 11
\][/tex]
### Conclusion
The equation of the median from vertex [tex]\( J \)[/tex] for the triangle with vertices [tex]\( J(2, 5) \)[/tex], [tex]\( K(4, -1) \)[/tex], and [tex]\( L(-2, -5) \)[/tex] is:
[tex]\[
y = 8x - 11
\][/tex]
