Given the circle below with chords start overline, V, W, end overline 
VW
  and start overline, X, Y, end overline 
XY
 . Find the length of start overline, X, Z, end overline 
XZ
 . Round to the nearest tenth if necessary.
Z
16
13
9
V
W
X
Y

This problem involves a circle with two intersecting chords \( VW \) and \( XY \). To find the length of \( XZ \), we can use the *chord-chord theorem*, which states that if two chords in a circle intersect, the products of the segments of each chord are equal.

Given:
- \( VZ = 16 \)
- \( ZW = 13 \)
- \( XZ = 9 \)
- We need to find \( ZY \).

### Step-by-Step Solution:
1. According to the chord-chord theorem:
   \[
   VZ \times ZW = XZ \times ZY
   \]
   Plugging in the known values:
   \[
   16 \times 13 = 9 \times ZY
   \]
   Simplify the left side:
   \[
   208 = 9 \times ZY
   \]
2. Solve for \( ZY \):
   \[
   ZY = \frac{208}{9} \approx 23.11
   \]

Thus, the length of \( ZY \) is approximately \( 23.1 \) when rounded to the nearest tenth.

Would you like further details or have any questions? Here are 5 related questions you might explore:

1. How does the chord-chord theorem apply to different shapes?
2. What if the circle had no intersecting chords—how would the geometry change?
3. Can we use the same theorem for tangents and secants in a circle?
4. What is the relationship between the radius and the chords in a circle?
5. How do you find the center of a circle when given two intersecting chords?

**Tip:** When working with circle theorems, always consider the relationships between intersecting lines, such as chords, tangents, or secants.

Given the circle below with chords VW and XY. Find the length of XZ. Round to the nearest tenth if necessary.

This problem involves finding the length of a chord in a circle using the chord-chord theorem. Given two intersecting chords, the product of the segments of one chord is equal to the product of the segments of the other. Learn how to solve this step-by-step using algebra and geometry. Suitable for Grades 9-11.

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