Understanding the Problem
To find the length of the chord of the outer circle that touches the inner circle, we can use geometric principles involving circles and right triangles.
Given Information
- Radius of the outer circle (R) = 13 cm
- Radius of the inner circle (r) = 5 cm
Visual Representation
- Imagine two concentric circles centered at the same point. The inner circle is completely inside the outer circle.
Using Geometry
1. Identify the Components:
- The distance from the center (O) to the point where the chord touches the inner circle (A) is equal to the radius of the inner circle (5 cm).
- The distance from the center (O) to the endpoints of the chord (B and C) is equal to the radius of the outer circle (13 cm).
2. Right Triangle Formation:
- Draw a radius from the center O to the midpoint (M) of the chord BC. This forms a right triangle OMA.
- Here, OA = 5 cm (radius of the inner circle) and OM = 13 cm (radius of the outer circle).
3. Calculate OM:
- The length of the segment AM can be calculated using the Pythagorean theorem:
- OM² = OA² + AM²
- 13² = 5² + AM²
- 169 = 25 + AM²
- AM² = 144
- AM = 12 cm
4. Find the Length of the Chord:
- Since M is the midpoint, the full length of the chord BC is:
- BC = 2 * AM = 2 * 12 cm = 24 cm
Final Answer
The length of the chord of the outer circle that touches the inner circle is 24 cm.