Questions: A line segment in the complex plane has endpoints 3+3i and -9-4i. Find its midpoint. Write your answer in the form a + bi. Simplify all fractions.

Solution Steps

The endpoints of the line segment in the complex plane are given as \(3 + 3i\) and \(-9 - 4i\).

The formula for the midpoint of a line segment with endpoints \(z_1 = a + bi\) and \(z_2 = c + di\) in the complex plane is:

\[ \text{Midpoint} = \left( \frac{a+c}{2} \right) + \left( \frac{b+d}{2} \right)i \]

Substitute the given endpoints into the midpoint formula:

\[ \text{Midpoint} = \left( \frac{3 + (-9)}{2} \right) + \left( \frac{3 + (-4)}{2} \right)i \]

Calculate the real and imaginary parts separately:

\[ \text{Real part} = \frac{3 - 9}{2} = \frac{-6}{2} = -3 \]

\[ \text{Imaginary part} = \frac{3 - 4}{2} = \frac{-1}{2} = -0.5 \]

Final Answer