Find the midpoint between
[tex]
\begin{array}{c}
8+11 i \
\text { and } \
-4-7 i \
{[?]+\square i}
\end{array}
[/tex]
Answer (2)
The midpoint between the complex numbers 8 + 11 i and − 4 − 7 i is found by averaging both their real and imaginary components, resulting in 2 + 2 i .
;
Find the sum of the real parts: 8 + ( − 4 ) = 4 .
Divide the sum of the real parts by 2: 2 4 = 2 .
Find the sum of the imaginary parts: 11 + ( − 7 ) = 4 .
Divide the sum of the imaginary parts by 2: 2 4 = 2 . The midpoint is 2 + 2 i .
Explanation
Understanding the Problem We are given two complex numbers, 8 + 11 i and − 4 − 7 i , and we want to find the midpoint between them.
Midpoint Formula The midpoint between two complex numbers z 1 = a + bi and z 2 = c + d i is found by averaging their real and imaginary parts: 2 z 1 + z 2 = 2 ( a + c ) + ( b + d ) i = 2 a + c + 2 b + d i .
Calculating the Real Part Let z 1 = 8 + 11 i and z 2 = − 4 − 7 i . We need to calculate the real part of the midpoint, which is 2 8 + ( − 4 ) .
Real Part Calculation The sum of the real parts is 8 + ( − 4 ) = 4 . Then, we divide by 2 to get the real part of the midpoint: 2 4 = 2 .
Calculating the Imaginary Part Next, we calculate the imaginary part of the midpoint, which is 2 11 + ( − 7 ) .
Imaginary Part Calculation The sum of the imaginary parts is 11 + ( − 7 ) = 4 . Then, we divide by 2 to get the imaginary part of the midpoint: 2 4 = 2 .
Final Answer Therefore, the midpoint is 2 + 2 i .
Examples
Complex numbers are used in electrical engineering to represent alternating current (AC) circuits. The midpoint formula can be used to find the average impedance of two circuits connected in series. This helps engineers design and analyze complex electrical systems. For example, if one circuit has an impedance of 8 + 11 i ohms and another has an impedance of − 4 − 7 i ohms, the average impedance is 2 + 2 i ohms.
