https://file.aiquickdraw.com/mathbot/file/17262331408485zwmcl1q.jpg

Let's analyze the given information from the problem.

- \( \overline{ZY} \cong \overline{XW} \), which means the lengths of segments \( ZY \) and \( XW \) are equal.
- \( ZX = 5x + 17 \)
- \( YW = 10 - 2x \)
- \( YX = 3 \)

We are asked to find the lengths of \( ZY \) and \( XW \), knowing that \( ZY = XW \).

### Step-by-step approach:

1. Since \( ZY \cong XW \), we know that \( ZY = XW \). The length of \( ZY \) can be expressed as the difference between \( ZX \) and \( XY \). So:
   \[
   ZY = ZX - XY = (5x + 17) - 3 = 5x + 14
   \]
2. Similarly, the length of \( XW \) can be expressed as the difference between \( YW \) and \( XY \):
   \[
   XW = YW - XY = (10 - 2x) - 3 = 7 - 2x
   \]
3. Since \( ZY = XW \), we can set the two equations equal to each other:
   \[
   5x + 14 = 7 - 2x
   \]
4. Solve for \( x \):
   \[
   5x + 2x = 7 - 14
   \]
   \[
   7x = -7
   \]
   \[
   x = -1
   \]
5. Now substitute \( x = -1 \) back into the expressions for \( ZY \) and \( XW \) to find their lengths:
   \[
   ZY = 5(-1) + 14 = -5 + 14 = 9
   \]
   \[
   XW = 7 - 2(-1) = 7 + 2 = 9
   \]

Therefore, both \( ZY \) and \( XW \) have lengths of 9 units.

### Final Answer:
\[
ZY = XW = 9
\]

Would you like a further explanation or details on any step?

Here are five related questions to deepen your understanding:
1. How would the problem change if \( ZX \) was a quadratic function of \( x \)?
2. Can you find the value of \( YW \) using the value of \( x \)?
3. What would happen if \( \overline{ZY} \) and \( \overline{XW} \) were not congruent?
4. How does the equality of segment lengths \( \overline{ZY} \cong \overline{XW} \) help solve this problem?
5. How would you graph the relationships between these segments on a coordinate plane?

**Tip**: Always double-check if geometric figures suggest congruence or equality in length; it can simplify solving equations in geometry problems.

In the figure, ZY ≅ XW, ZX = 5x + 17, YW = 10 - 2x, and YX = 3. Find ZY and XW.

Learn how to solve a geometry problem involving congruent segments. This example explores finding the lengths of ZY and XW using algebraic equations with given values for ZX, YW, and YX. Suitable for students in Grades 9-10.

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