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To solve this problem, we need to determine the length of the chord \( YZ \) in the circle, where the radius of the circle is given as 7 inches, and the center of the circle is \( X \).

### Key Points:
- The radius \( XY = XZ = 7 \) inches.
- \( X \) is the center of the circle.
- \( YZ \) is a chord of the circle.

### Steps to Solve:
1. **Identify Triangle \( XYZ \)**:
   - Triangle \( XYZ \) is an isosceles triangle because \( XY = XZ = 7 \) inches.

2. **Use the Perpendicular from the Center**:
   - The perpendicular from the center \( X \) to the chord \( YZ \) bisects the chord.
   - Let the midpoint of \( YZ \) be \( M \). Then, \( XM \) is perpendicular to \( YZ \), and \( YM = MZ \).

3. **Apply the Pythagorean Theorem**:
   - Let \( XM = h \) and \( YM = MZ = \frac{YZ}{2} \).
   - Using the Pythagorean theorem in \( \triangle XMY \):
     \[
     XY^2 = XM^2 + YM^2
     \]
     \[
     7^2 = h^2 + \left(\frac{YZ}{2}\right)^2
     \]
     \[
     49 = h^2 + \frac{(YZ)^2}{4}
     \]

4. **Given \( h = 0 \)** for Diameter**:
   - If \( YZ \) passes through \( X \), then \( h = 0 \).
   - Therefore:
     \[
     49 = \frac{(YZ)^2}{4}
     \]
     \[
     (YZ)^2 = 49 \times 4
     \]
     \[
     (YZ)^2 = 196
     \]
     \[
     YZ = \sqrt{196} = 14 \text{ inches}
     \]

Thus, the length of \( YZ \) is **14 inches**.

Would you like further details or have any questions? Here are some related questions to explore:

1. How would the length of \( YZ \) change if the radius were different?
2. What is the relationship between the chord length and the distance from the center?
3. Can the length of any chord in a circle be longer than the diameter?
4. How would the equation change if the chord did not pass through the center?
5. What other methods could be used to find the length of a chord?

**Tip:** Always remember that a chord passing through the center of a circle is a diameter, and it's the longest chord in the circle.

The radius of the circle shown below is 7 inches. The center is X. The length of YZ is ____ inches.

Learn how to calculate the length of a chord in a circle with a 7-inch radius. This problem involves geometric concepts, including the Pythagorean Theorem and the Perpendicular Bisector Theorem. Suitable for students in Grades 8-10.

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