To determine the coordinates of the other endpoint of the line segment, we use the given midpoint formula and the information provided.
Let's denote the two endpoints of the line segment as [tex]\( Z(x_1, y_1) \)[/tex] and [tex]\( A(x_2, y_2) \)[/tex]. The midpoint [tex]\( M \)[/tex] of a line segment with endpoints [tex]\( Z \)[/tex] and [tex]\( A \)[/tex] is given by the following formulas:
[tex]\[
M_x = \frac{x_1 + x_2}{2}, \quad M_y = \frac{y_1 + y_2}{2}
\][/tex]
Given:
- One endpoint [tex]\( Z(-21, 15) \)[/tex]
- Midpoint [tex]\( M(-13, 29) \)[/tex]
We need to find the coordinates of the other endpoint [tex]\( A(x_2, y_2) \)[/tex].
From the midpoint formulas, we know:
[tex]\[
-13 = \frac{-21 + x_2}{2} \quad (1)
\][/tex]
[tex]\[
29 = \frac{15 + y_2}{2} \quad (2)
\][/tex]
We can solve these equations step-by-step:
1. Solve for [tex]\( x_2 \)[/tex]:
[tex]\[
-13 = \frac{-21 + x_2}{2}
\][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[
-26 = -21 + x_2
\][/tex]
Add 21 to both sides to solve for [tex]\( x_2 \)[/tex]:
[tex]\[
x_2 = -5
\][/tex]
2. Solve for [tex]\( y_2 \)[/tex]:
[tex]\[
29 = \frac{15 + y_2}{2}
\][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[
58 = 15 + y_2
\][/tex]
Subtract 15 from both sides to solve for [tex]\( y_2 \)[/tex]:
[tex]\[
y_2 = 43
\][/tex]
Therefore, the coordinates of the other endpoint [tex]\( A \)[/tex] are [tex]\( (-5, 43) \)[/tex].
So, the endpoint of the line segment is:
[tex]\[
\boxed{(-5, 43)}
\][/tex]
