What is the length of AZAZ? Use the following information: 

*AA* is between *YY* and *ZZ*, *YA=24YA=24*, AZ = 15xAZ = 15x, and YZ = 189YZ = 189.

To solve for the length of \( AZ \), let's break down the information step by step:

### Given Information:
1. **\( AA \)** is between **\( YY \)** and **\( ZZ \)** (This suggests that the points \( A \), \( Y \), and \( Z \) are collinear).
2. \( YA = 24 \)
3. \( AZ = 15x \)
4. \( YZ = 189 \)

### Goal:
We need to find the length of \( AZ \).

### Step 1: Relationship between \( Y \), \( A \), and \( Z \)
From the information, since \( YA = 24 \) and \( YZ = 189 \), we can deduce the distance between the points.

### Step 2: Expressing \( YZ \) in terms of known distances
The total length \( YZ \) is the sum of the distances \( YA \) and \( AZ \), so:

\[
YZ = YA + AZ
\]

Substituting the known values:

\[
189 = 24 + AZ
\]

### Step 3: Solving for \( AZ \)
Solving for \( AZ \) from the above equation:

\[
AZ = 189 - 24 = 165
\]

Thus, the length of \( AZ \) is 165 units.

Would you like more details or have any questions?

Here are 5 related questions:
1. What if \( YA = 30 \) and \( YZ = 200 \)? What would be the length of \( AZ \)?
2. How would the result change if the total length \( YZ \) were shorter than 189?
3. What if \( AZ \) was described as a function of \( x \) instead of a constant?
4. What happens to the lengths if the positions of \( Y \), \( A \), and \( Z \) are reversed?
5. How can this be applied to solving distance problems in coordinate geometry?

**Tip:** When dealing with line segments, ensure you correctly add or subtract the distances between points based on their relative positions on the line.

Learn how to calculate the length of AZ using coordinate geometry principles. Given points A, Y, and Z are collinear with specified distances YA = 24, AZ = 15x, and YZ = 189, find the length of AZ step-by-step. Suitable for high school students learning coordinate geometry.

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