In a circle with center O and radius 5, segment O C is perpendicular to segment A B at C. A and B lie on the circle and A B equals 6. What is the length of segment O C?

Question

Answer options with 5 options
A.
8

B.
6

C.
5

D.
4

E.
3

Answer 1

To solve for the length of segment \( OC \), we can start by analyzing the geometric situation.

1. Given Information:

   • The radius of the circle is \( r = 5 \).
   • \( A \) and \( B \) are points on the circle such that \( AB = 6 \).
   • \( OC \) is perpendicular to \( AB \) at point \( C \).

2. Properties of the Circle:

   • Since \( A \) and \( B \) are points on the circle, the distance from the center \( O \) to either point \( A \) or \( B \) is equal to the radius, which is \( 5 \).

3. Positioning:

   • Place the center \( O \) at the origin \( (0, 0) \).
   • Let \( C \) be at coordinate \( (0, h) \), where \( h \) is the length of segment \( OC \).
   • Points \( A \) and \( B \) will lie on the horizontal line through \( C \). Since \( AB = 6 \), the coordinates of \( A \) and \( B \) can be expressed as \( A(-3, h) \) and \( B(3, h) \).

4. Using Pythagoras’ theorem:

   • Using the distance from the center \( O(0, 0) \) to point \( A(-3, h) \), we apply the Pythagorean theorem: \[ OA = \sqrt{(-3)^2 + h^2} = 5. \]
   • Squaring both sides gives: \[ 9 + h^2 = 25. \]
   • Solving for \( h^2 \): \[ h^2 = 25 - 9 = 16. \]
   • Therefore, taking the square root: \[ h = 4. \]

5. Conclusion:

   • The length of segment \( OC \) is \( \boxed{4} \).