Video Transcript
What complex number lies at the midpoint of 𝑧 one and 𝑧 two on the given complex plane?
We know that any point on an Argand diagram with Cartesian coordinates 𝑥, 𝑦 can be written as a complex number 𝑧 is equal to 𝑥 plus 𝑦𝑖. The 𝑥-coordinate is the real part, and the 𝑦-coordinate is the imaginary part. 𝑧 sub one has coordinates negative two, seven. This means that it represents the complex number negative two plus seven 𝑖. 𝑧 sub two has coordinates six, negative three. This means that 𝑧 two is equal to the complex number six minus three 𝑖. We need to find the midpoint of these two complex numbers.
We know that the midpoint of any two coordinates has 𝑥-coordinate 𝑥 one plus 𝑥 two divided by two and 𝑦-coordinate 𝑦 one plus 𝑦 two divided by two. We need to find the average or midpoint of the 𝑥- and 𝑦-coordinates separately. The 𝑥-coordinates are negative two and six, and the 𝑦-coordinates are seven and negative three. Negative two plus six is equal to four. And dividing this by two gives us two. In the same way, seven plus negative three is also equal to four. And dividing this by two also gives us two. The coordinates of the midpoint of 𝑧 one and 𝑧 two are two, two. This is equivalent to the complex number two plus two 𝑖.
We can also show this on the Argand diagram. To get from 𝑧 one to the midpoint, we go along four units and down five units. We also go along four units and down five units to get from the midpoint to 𝑧 two. This confirms that the midpoint has coordinates two, two and complex number two plus two 𝑖.
