To find the other endpoint of a segment in the complex plane when one endpoint and the midpoint are given, we use the midpoint formula.
The midpoint [tex]\( M \)[/tex] of a segment with endpoints [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is given by:
[tex]\[ M = \left( \frac{x_1 + x_2}{2} \, , \, \frac{y_1 + y_2}{2} \right) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] are the coordinates of endpoint [tex]\( A \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] are the coordinates of endpoint [tex]\( B \)[/tex].
Given:
- The midpoint [tex]\( M \)[/tex] is [tex]\( 7 - 2i \)[/tex]. So, the coordinates of the midpoint are [tex]\( (7, -2) \)[/tex].
- One endpoint (let's call it [tex]\( A \)[/tex]) is [tex]\( 11 - 3i \)[/tex]. So, the coordinates of endpoint [tex]\( A \)[/tex] are [tex]\( (11, -3) \)[/tex].
We need to find the coordinates of the other endpoint [tex]\( B \)[/tex] (let's call its coordinates [tex]\( (x_2, y_2) \)[/tex]).
Using the midpoint formula, we have:
[tex]\[ 7 = \frac{11 + x_2}{2} \][/tex]
[tex]\[ -2 = \frac{-3 + y_2}{2} \][/tex]
Let's solve these equations step-by-step.
1. Solve for [tex]\( x_2 \)[/tex]:
[tex]\[ 7 = \frac{11 + x_2}{2} \][/tex]
Multiply both sides by 2:
[tex]\[ 14 = 11 + x_2 \][/tex]
Subtract 11 from both sides:
[tex]\[ 14 - 11 = x_2 \][/tex]
[tex]\[ x_2 = 3 \][/tex]
2. Solve for [tex]\( y_2 \)[/tex]:
[tex]\[ -2 = \frac{-3 + y_2}{2} \][/tex]
Multiply both sides by 2:
[tex]\[ -4 = -3 + y_2 \][/tex]
Add 3 to both sides:
[tex]\[ -4 + 3 = y_2 \][/tex]
[tex]\[ y_2 = -1 \][/tex]
Thus, the coordinates of the other endpoint [tex]\( B \)[/tex] are [tex]\( (3, -1) \)[/tex], and in the complex form, it is [tex]\( 3 - i \)[/tex].
Hence, the other endpoint is:
[tex]\[ \boxed{3 - i} \][/tex]
