Determinethe length of the radius of the circle.

To determine the length of the radius of the circle, we need to use the information given in the problem and apply the Pythagorean theorem.

### Given:
- \( AB = 36 \) cm (Chord length)
- \( D \) is the midpoint of \( AB \), so \( AD = DB = \frac{AB}{2} = 18 \) cm.
- \( OD = x \) is the perpendicular distance from the center \( O \) to the chord \( AB \).

### Steps to Find the Radius:

We know that the radius \( r \) of the circle is \( OA = OB = r \). Let's focus on the right triangle \( OAD \):

\[
OA^2 = OD^2 + AD^2
\]

Given that \( AD = 18 \) cm, the equation becomes:

\[
r^2 = x^2 + 18^2.
\]

Since we want to find the radius \( r \), we need to know the value of \( x \), the perpendicular distance from \( O \) to the chord \( AB \). However, the problem does not explicitly provide \( x \), but the radius \( r \) can be determined if \( OD = x \) is known.

### Applying the Pythagorean Theorem:

Let us assume the standard equation:

\[
r^2 = x^2 + 18^2.
\]

We can rearrange this equation to solve for \( r \):

\[
r = \sqrt{x^2 + 18^2}.
\]

### Finding \( x \):

If the problem's instructions require that \( OD = x \) remains undetermined, we can't solve for \( r \) numerically without additional information. However, if it is assumed \( x = 12 \) cm (for example), the calculation becomes:

\[
r^2 = 12^2 + 18^2 = 144 + 324 = 468,
\]

\[
r = \sqrt{468} \approx 21.63 \, \text{cm}.
\]

To determine the exact value, please confirm if any additional information such as the specific \( x \) value is available or another parameter.

Would you like to provide the value of \( x \), or ask for additional information on this process?

### Follow-up Questions:

1. How would the radius change if the perpendicular distance from the chord to the center \( x \) was different?
2. What geometric properties determine the perpendicular distance \( x \) to any chord?
3. If the chord length changes, how does it affect the radius and center positioning?
4. What other geometric properties can be used to find \( x \) if it's missing?
5. How does the angle formed at the circle's center influence the chord's length?

### Tip:
When solving geometry problems involving circles and chords, always consider the properties of right triangles and perpendicular bisectors to understand the relationships between various components in the circle.

Learn how to determine the length of the radius of a circle using the Pythagorean theorem. Understand the relationship between the radius, chord length, and perpendicular distance from the center to the chord. Suitable for high school students studying geometry and circle properties.

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