Two concentric circles with radius 3 cm and 9.25 cm. Find the length of the chord of the bigger circle which is tangent to the other circle.

Options

A. 17.5 cm
B. 19.25 cm
C. 12.5 cm
D. 15 cm

Answer

To find the length of the chord of the bigger circle that is tangent to the smaller circle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Radii of the Circles:**
- The radius of the smaller circle (r1) = 3 cm.
- The radius of the bigger circle (r2) = 9.25 cm.

2. **Understand the Geometry:**
- The two circles are concentric, meaning they share the same center (let's denote this center as point O).
- The chord of the bigger circle (let's denote the endpoints of the chord as A and B) is tangent to the smaller circle.

3. **Draw the Right Triangle:**
- Since the chord AB is tangent to the smaller circle at point C, the radius OC (which is the radius of the smaller circle) is perpendicular to the chord AB.
- This creates a right triangle OAC, where:
- OA = r2 = 9.25 cm (the radius of the bigger circle),
- OC = r1 = 3 cm (the radius of the smaller circle),
- AC is half the length of the chord AB.

4. **Apply the Pythagorean Theorem:**
- In triangle OAC, we can use the Pythagorean theorem:
\[ OA^2 = OC^2 + AC^2 \]
- Substituting the known values:
\[ (9.25)^2 = (3)^2 + AC^2 \]
- This simplifies to:
\[ 85.5625 = 9 + AC^2 \]
- Rearranging gives:
\[ AC^2 = 85.5625 - 9 = 76.5625 \]

5. **Calculate AC:**
- Taking the square root of both sides:
\[ AC = \sqrt{76.5625} = 8.75 \text{ cm} \]

6. **Find the Length of the Chord AB:**
- Since AC is half of the chord AB, we have:
\[ AB = AC + BC = 8.75 + 8.75 = 17.5 \text{ cm} \]

### Final Answer:
The length of the chord AB of the bigger circle that is tangent to the smaller circle is **17.5 cm**.