To solve the problem of finding the coordinates of the other endpoint of the line segment, given the midpoint and one endpoint, we use the midpoint formula.
The midpoint formula for a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
In this problem, we are given:
- The midpoint [tex]\(M = (1, 3)\)[/tex]
- One endpoint [tex]\((x_1, y_1) = (-8, 8)\)[/tex]
We need to find the coordinates of the other endpoint [tex]\((x_2, y_2)\)[/tex].
Let's set up the equations using the midpoint formula:
Since the midpoint's x-coordinate is 1:
[tex]\[
1 = \frac{-8 + x_2}{2}
\][/tex]
Similarly, since the midpoint's y-coordinate is 3:
[tex]\[
3 = \frac{8 + y_2}{2}
\][/tex]
We will solve these equations step-by-step:
1. Solve for [tex]\(x_2\)[/tex]:
[tex]\[
1 = \frac{-8 + x_2}{2}
\][/tex]
Multiply both sides by 2:
[tex]\[
2 = -8 + x_2
\][/tex]
Add 8 to both sides:
[tex]\[
x_2 = 2 + 8
\][/tex]
[tex]\[
x_2 = 10
\][/tex]
2. Solve for [tex]\(y_2\)[/tex]:
[tex]\[
3 = \frac{8 + y_2}{2}
\][/tex]
Multiply both sides by 2:
[tex]\[
6 = 8 + y_2
\][/tex]
Subtract 8 from both sides:
[tex]\[
y_2 = 6 - 8
\][/tex]
[tex]\[
y_2 = -2
\][/tex]
Thus, the coordinates of the other endpoint are [tex]\((10, -2)\)[/tex].
