To determine the other endpoint of a segment in the complex plane, given one endpoint and the midpoint, we use the midpoint formula for complex numbers. Let's work through the steps to find this endpoint.
Given:
- Midpoint: [tex]\( 3 - 2i \)[/tex]
- One endpoint: [tex]\( 7 + i \)[/tex]
Let the unknown second endpoint be [tex]\( z_2 \)[/tex] which can be represented as [tex]\( x + yi \)[/tex].
The midpoint formula in the complex plane states that the real part and the imaginary part of the midpoint are the averages of the real parts and the imaginary parts of the endpoints, respectively. Mathematically, if the midpoint [tex]\( M \)[/tex] is given by [tex]\( \left( \frac{a+b}{2}, \frac{c+d}{2} \right) \)[/tex] where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the real parts, and [tex]\( c \)[/tex] and [tex]\( d \)[/tex] are the imaginary parts of the endpoints, respectively, we can set up the following equations:
For the real part:
[tex]\[ \frac{7 + x}{2} = 3 \][/tex]
For the imaginary part:
[tex]\[ \frac{1 + y}{2} = -2 \][/tex]
Now, solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
1. Solve the real part equation:
[tex]\[ \frac{7 + x}{2} = 3 \][/tex]
Multiply both sides by 2:
[tex]\[ 7 + x = 6 \][/tex]
Subtract 7 from both sides:
[tex]\[ x = -1 \][/tex]
2. Solve the imaginary part equation:
[tex]\[ \frac{1 + y}{2} = -2 \][/tex]
Multiply both sides by 2:
[tex]\[ 1 + y = -4 \][/tex]
Subtract 1 from both sides:
[tex]\[ y = -5 \][/tex]
Therefore, the other endpoint is [tex]\( x + yi = -1 - 5i \)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{-1 - 5i} \][/tex]
