Two equal circles pass through the center of each other. If the radius of each circle is 5 cm, then the length of their common chord is :

Question

Accepted Answer

To determine the length of the common chord of two equal circles that pass through each other's centers, we can use geometric reasoning and the properties of circles.

### Explanation:

1. **Understanding the Configuration:**
   - Each circle has a radius of 5 cm.
   - The centers of the two circles are each located on the circumference of the other circle. This means the distance between the centers of the two circles is equal to the radius, which is 5 cm.

2. **Geometry of the Situation:**
   - The common chord is the line segment that is equidistant from the centers of both circles and perpendicular to the line joining the centers.
   - Since the circles are equal and pass through each other's centers, the line joining the centers is the perpendicular bisector of the common chord.

3. **Using the Right Triangle:**
   - Consider the triangle formed by the center of one circle (say, O), the midpoint of the common chord (M), and one endpoint of the chord (A).
   - This triangle OMA is a right triangle, where OM is the perpendicular from the center to the chord, and OA is the radius of the circle.
   - The length of OA is 5 cm (radius of the circle), and OM is half the distance between the centers, which is 2.5 cm (since the full distance is 5 cm).

4. **Applying the Pythagorean Theorem:**
   - In the right triangle OMA, we have:
     $$
     OA^2 = OM^2 + AM^2
     $$
     $$
     5^2 = 2.5^2 + AM^2
     $$
     $$
     25 = 6.25 + AM^2
     $$
     $$
     AM^2 = 18.75
     $$
     $$
     AM = \sqrt{18.75} = \frac{5\sqrt{3}}{2}
     $$

5. **Length of the Common Chord:**
   - Since AM is half of the chord, the full length of the chord (AB) is twice AM:
     $$
     AB = 2 \times \frac{5\sqrt{3}}{2} = 5\sqrt{3}
     $$

### Conclusion:

- **Correct Option: 2. 5$\sqrt{3}$**
  - This matches the calculated length of the common chord, confirming that Option 2 is correct.

### Why Other Options are Incorrect:

1. **Option 1: 5**
   - This option suggests the chord length is equal to the radius, which is incorrect because the chord is longer than the radius due to the geometry of the circles.

2. **Option 3: 10$\sqrt{3}$**
   - This option suggests a length that is too long. The calculation shows the chord is 5$\sqrt{3}$, not 10$\sqrt{3}$.

3. **Option 4: $\frac{5\sqrt{3}}{2}$**
   - This option represents half the length of the chord, not the full length. The full length is 5$\sqrt{3}$.

Thus, the correct answer is indeed Option 2: 5$\sqrt{3}$.