Out of two concentric circles, the diameter of the outer circle is 26 cm and the chord MN of length 24 cm is tangent to the inner circle. The radius of the inner circle is :

Question

Accepted Answer

To determine the radius of the inner circle, we need to analyze the given information about the concentric circles and the chord MN.

1. **Understanding the Problem:**
   - We have two concentric circles, meaning they share the same center.
   - The diameter of the outer circle is 26 cm, so its radius (R) is 13 cm.
   - The chord MN of the outer circle is 24 cm long and is tangent to the inner circle.

2. **Geometric Configuration:**
   - Since MN is tangent to the inner circle, the perpendicular from the center of the circles (let's call it O) to the chord MN will also be the radius of the inner circle (r).
   - Let P be the point where this perpendicular meets MN. Since MN is a chord, OP is perpendicular to MN and bisects it. Therefore, MP = PN = 12 cm.

3. **Applying the Pythagorean Theorem:**
   - In the right triangle OMP, where OM is the radius of the outer circle (13 cm), OP is the radius of the inner circle (r), and MP is half the chord length (12 cm), we can apply the Pythagorean theorem:
     $$
     OM^2 = OP^2 + MP^2
     $$
     $$
     13^2 = r^2 + 12^2
     $$
     $$
     169 = r^2 + 144
     $$
     $$
     r^2 = 169 - 144
     $$
     $$
     r^2 = 25
     $$
     $$
     r = 5 \text{ cm}
     $$

4. **Conclusion:**
   - The radius of the inner circle is 5 cm, which corresponds to Option 1.

5. **Explanation of Incorrect Options:**
   - **Option 2 (6 cm):** If the radius were 6 cm, then using the Pythagorean theorem, the calculation would not satisfy the equation derived from the given dimensions.
   - **Option 3 (8 cm):** Similarly, a radius of 8 cm would not satisfy the Pythagorean relationship with the given chord length and outer circle radius.
   - **Option 4 (10 cm):** A radius of 10 cm would exceed the possible length derived from the Pythagorean theorem, given the constraints of the problem.

Thus, the correct answer is indeed Option 1, with the radius of the inner circle being 5 cm.