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.