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We are given two complex numbers, \( z_1 \) and \( z_2 \), represented as points on the complex plane. Let's start by determining their values and then calculate the distance between them.

### Step 1: Identify the coordinates of \( z_1 \) and \( z_2 \)

From the graph:
- \( z_1 \) is located at (1, -4), which corresponds to the complex number \( z_1 = 1 - 4i \).
- \( z_2 \) is located at (3, 9), which corresponds to the complex number \( z_2 = 3 + 9i \).

### Step 2: Calculate the distance between \( z_1 \) and \( z_2 \)

The formula to find the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the complex plane is:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substitute the coordinates of \( z_1 \) and \( z_2 \):
\[
\text{Distance} = \sqrt{(3 - 1)^2 + (9 - (-4))^2}
= \sqrt{(2)^2 + (13)^2}
= \sqrt{4 + 169}
= \sqrt{173}
\]
Thus, the distance between \( z_1 \) and \( z_2 \) is \( \sqrt{173} \).

### Final Answer:
- \( z_1 = 1 - 4i \)
- \( z_2 = 3 + 9i \)
- Distance = \( \sqrt{173} \)

Would you like a detailed breakdown of these steps or have any other questions? Here are some related questions to explore:

1. How do you interpret the real and imaginary parts of a complex number on the complex plane?
2. What is the geometric meaning of the distance formula in the context of complex numbers?
3. How would the distance change if the points were located in different quadrants?
4. Can you represent the distance between two complex numbers as a vector?
5. How do operations such as addition and subtraction work geometrically with complex numbers?

**Tip:** When working with complex numbers, always consider their geometric interpretation to simplify understanding operations like addition, multiplication, and finding distances.

Find the complex numbers z1 and z2. Then, find the distance between them.

Learn how to find the complex numbers z1 and z2 from the graph and calculate the distance between them. This problem involves key concepts like complex numbers in the form a + bi and the distance formula in the complex plane. Suitable for students in Grades 10-12.

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