To determine the coordinates of the other endpoint [tex]\( K \)[/tex] given the midpoint [tex]\( L(-1, 8) \)[/tex] and one endpoint [tex]\( J(4, -15) \)[/tex], we need to use the midpoint formula. The midpoint formula states:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Where [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are the coordinates of the endpoints of the line segment, and the coordinates of the midpoint are [tex]\((x_m, y_m)\)[/tex].
From the given problem:
- Coordinates of midpoint [tex]\( L \)[/tex] are [tex]\((-1, 8)\)[/tex].
- Coordinates of endpoint [tex]\( J \)[/tex] are [tex]\((4, -15)\)[/tex].
We are to find the coordinates of the other endpoint [tex]\( K \)[/tex], say [tex]\((x_1, y_1)\)[/tex].
### Solve for the x-coordinate
The midpoint formula for the x-coordinates is:
[tex]\[
\frac{x_1 + x_2}{2} = x_m
\][/tex]
Plugging in the known values:
[tex]\[
\frac{4 + x_1}{2} = -1
\][/tex]
This is one of the equations, thus:
[tex]\[
\frac{4 + x_1}{2} = -1
\][/tex]
### Solve for the y-coordinate
The midpoint formula for the y-coordinates is:
[tex]\[
\frac{y_1 + y_2}{2} = y_m
\][/tex]
Plugging in the known values:
[tex]\[
\frac{-15 + y_1}{2} = 8
\][/tex]
This is another equation, thus:
[tex]\[
\frac{-15 + y_1}{2} = 8
\][/tex]
### Conclusion
The two equations that can be solved to determine the coordinates of the other endpoint [tex]\( K \)[/tex] are:
- [tex]\(\frac{4 + x_1}{2} = -1\)[/tex]
- [tex]\(\frac{-15 + y_1}{2} = 8\)[/tex]
These equations correctly use the midpoint formula and the given values to find the unknown endpoint [tex]\( K \)[/tex].
