How to Find Distance and Midpoint of Complex Numbers?

We all know how to calculate the distance between two spots and the midpoint between two points in a Cartesian plane, but what happens when we’re dealing with something more complicated like a complex plane?

It turns out that the formulae used to get the distance between two complex numbers and the midpoint between two complex numbers are very similar to the formulae used to determine the distance between two Cartesian points.

Complex numbers are discussed in this part, and we will learn how to utilize both the midpoint formula and the distance formula for them.

Related Topics

• How to Find the Distance of Two Points
• How to Find Midpoint
• How to Solve the Complex Plane

A step-by-step guide to finding the distance and midpoint of complex numbers

The following is the formula for calculating the distance between two points \((a, b)\) on the complex plane for the complex number \(a + bi\) and \((c, d)\) for the complex number \(c + di\):

\(\color{blue}{d=\sqrt{(c-a)^{2}+(d-b){^2}}}\)

• It makes no difference which points you deduct as far as the order remains the same. If you subtract the real component of the first integer from the second, you must also subtract the first number’s imaginary part from the second.

Using the midpoint rule from coordinate geometry, we can get the midpoint of any two complex numbers \(z=a+bi\) and \(w=c+di\):

\(\color{blue}{M=\frac{a+c}{2}+(\frac{b+d}{2})i}\)

• Thus, the complex number at the middle of the line is given.

Finding Distance and Midpoint of Complex Numbers – Example 1:

Find the distance between the points \(3+2i\) and \(8+5i\).

To find the distance between complex numbers use this formula: \(\color{blue}{d=\sqrt{(c-a)^{2}+(d-b){^2}}}\).

The distance is: \(d=\sqrt{(8-3)^2+(5i-2i)^2}=\sqrt{(5)^2+(3i)^2}\)

Then: \(i^2=-1\)

\(d=\sqrt{(5)^2+(3i)^2}=\sqrt{(5)^2+(3^2 i^2)}=\sqrt {(25)+(-9)}=\sqrt{16}=4\)

Finding Distance and Midpoint of Complex Numbers – Example 2:

Find the midpoint between the points \(6-4i\) and \(2+2i\)

To find the midpoint between complex numbers use this formula: \(\color{blue}{M=\frac{a+c}{2}+(\frac{b+d}{2})i}\)

\(M= \frac {6+2}{2}+(\frac {-4+2}{2})i\)\(=\)\( \frac {8}{2}+(\frac {-2}{2})i=4-i\)

Exercises for Finding Distance and Midpoint of Complex Numbers

Find the distance and midpoint of complex numbers.

1. \(\color{blue}{(1+2i)}\) , \(\color{blue}{(-2+4i)}\)
2. \(\color{blue}{(-6+2i)}\) , \(\color{blue}{(-2+5i)}\)
3. \(\color{blue}{(-7i)}\), \(\color{blue}{(5-4i)}\)
4. \(\color{blue}{(2+i)}\), \(\color{blue}{(6+5i)}\)
5. \(\color{blue}{(-7-3i)}\), \(\color{blue}{(3+5i)}\)

1. \(\color{blue}{d=\sqrt{5}}\), \(\color{blue}{M=-\frac{1}{2}+3i}\)
2. \(\color{blue}{d=\sqrt{7}}\), \(\color{blue}{M=-4+\frac{7}{2}i}\)
3. \(\color{blue}{d=4}\), \(\color{blue}{M=\frac{5}{2}-\frac{11}{2}i}\)
4. \(\color{blue}{d=0}\), \(\color{blue}{M=4+3i}\)
5. \(\color{blue}{d=6}\), \(\color{blue}{M=-2+i}\)