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We are given a circle with center \( O \) and radius 5. The line segment \( AB \) is a chord in the circle with length \( AB = 6 \), and \( OC \) is perpendicular to \( AB \) at point \( C \). We are tasked with finding the length of \( OC \).

### Solution:

Since \( OC \) is perpendicular to \( AB \) at point \( C \), it bisects the chord \( AB \). Therefore, the segments \( AC \) and \( BC \) are equal in length:
\[
AC = BC = \frac{AB}{2} = \frac{6}{2} = 3.
\]

Now, we can apply the Pythagorean Theorem to the right triangle \( OCA \) where:
- \( OA \) is the radius of the circle, \( OA = 5 \),
- \( AC = 3 \),
- \( OC \) is the perpendicular from the center \( O \) to the chord \( AB \).

By the Pythagorean Theorem:
\[
OA^2 = OC^2 + AC^2.
\]
Substituting the known values:
\[
5^2 = OC^2 + 3^2,
\]
\[
25 = OC^2 + 9,
\]
\[
OC^2 = 25 - 9 = 16,
\]
\[
OC = \sqrt{16} = 4.
\]

Thus, the length of \( OC \) is \( \boxed{4} \).

Would you like more details or have any other questions?

Here are five related questions to expand on this problem:
1. How do you find the distance from the center of a circle to a chord if the radius and chord length are given?
2. What is the formula to find the length of a perpendicular from the center of a circle to a chord?
3. How does the Pythagorean Theorem apply to right triangles in a circle?
4. How do you calculate the radius of a circle given the distance to a chord and the length of the chord?
5. Can the perpendicular from the center of a circle always bisect any chord?

**Tip:** When you have a right triangle in a circle, it's often useful to apply the Pythagorean Theorem by using the radius, chord, and the perpendicular from the center.

In a circle with center O and radius 5, OC ⊥ AB at C. A and B lie on the circle and AB = 6. What is the length of OC?

Learn how to find the length of the perpendicular from the center of a circle to a chord using the Pythagorean theorem. Given a circle with radius 5 and a chord of length 6, this solution shows how to calculate OC. Suitable for Grades 9-11.

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