To determine the coordinates of the other endpoint of a line segment when one endpoint and the midpoint are given, we can utilize the midpoint formula. The midpoint formula states that the coordinates of the midpoint [tex]\((xm, ym)\)[/tex] of a line segment with endpoints [tex]\((x1, y1)\)[/tex] and [tex]\((x2, y2)\)[/tex] are given by:
[tex]\[
xm = \frac{x1 + x2}{2}, \quad ym = \frac{y1 + y2}{2}
\][/tex]
We are given:
- One endpoint [tex]\((x1, y1) = (-2, 5)\)[/tex]
- The midpoint [tex]\((xm, ym) = (3, 1)\)[/tex]
We need to find the coordinates of the other endpoint [tex]\((x2, y2)\)[/tex].
### Step-by-Step Solution
1. Express the midpoint coordinates in terms of the endpoints:
[tex]\[
xm = \frac{x1 + x2}{2}, \quad ym = \frac{y1 + y2}{2}
\][/tex]
2. Substitute the known coordinates of the midpoint and one endpoint into the equations:
[tex]\[
3 = \frac{-2 + x2}{2}, \quad 1 = \frac{5 + y2}{2}
\][/tex]
3. Solve for [tex]\(x2\)[/tex]:
[tex]\[
3 = \frac{-2 + x2}{2}
\][/tex]
Multiply both sides by 2:
[tex]\[
6 = -2 + x2
\][/tex]
Add 2 to both sides:
[tex]\[
8 = x2
\][/tex]
So, [tex]\(x2 = 8\)[/tex].
4. Solve for [tex]\(y2\)[/tex]:
[tex]\[
1 = \frac{5 + y2}{2}
\][/tex]
Multiply both sides by 2:
[tex]\[
2 = 5 + y2
\][/tex]
Subtract 5 from both sides:
[tex]\[
-3 = y2
\][/tex]
So, [tex]\(y2 = -3\)[/tex].
5. Thus, the coordinates of the other endpoint are:
[tex]\[
(x2, y2) = (8, -3)
\][/tex]
After verifying our work with the calculations, the correct answer is:
[tex]\[
\boxed{(8, -3)}
\][/tex]
