Two equal circles of radius r intersect such that each passes through the centre of the other. The length of the common chord of the circles is :

Question

Accepted Answer

To determine the length of the common chord of two equal circles of radius $ r $ that intersect such that each passes through the center of the other, we can use geometric reasoning and properties of circles.

### Explanation Supporting Option 4:

1. **Geometric Configuration**:
   - Consider two circles, Circle A and Circle B, each with radius $ r $.
   - The center of Circle A is at point O, and the center of Circle B is at point O'.
   - Since each circle passes through the center of the other, the distance between O and O' is $ r $.

2. **Formation of an Equilateral Triangle**:
   - The points O, O', and the midpoint M of the common chord form an equilateral triangle.
   - This is because OO' = OM = O'M = $ r $.

3. **Using the Pythagorean Theorem**:
   - The common chord is perpendicular to the line segment OO' at its midpoint M.
   - Let the length of the common chord be $ AB $.
   - In the right triangle OMA, where OM = $ \frac{r}{2} $ (since M is the midpoint of OO'), and OA = $ r $, we apply the Pythagorean theorem:
     $$
     OA^2 = OM^2 + AM^2
     $$
     $$
     r^2 = \left(\frac{r}{2}\right)^2 + AM^2
     $$
     $$
     r^2 = \frac{r^2}{4} + AM^2
     $$
     $$
     AM^2 = r^2 - \frac{r^2}{4} = \frac{3r^2}{4}
     $$
     $$
     AM = \frac{\sqrt{3}r}{2}
     $$

4. **Length of the Common Chord**:
   - Since M is the midpoint of the chord AB, the full length of the chord is:
     $$
     AB = 2 \times AM = 2 \times \frac{\sqrt{3}r}{2} = \sqrt{3}r
     $$

Thus, the length of the common chord is $\sqrt{3}r$, which confirms that Option 4 is correct.

### Explanation for Why Other Options are Incorrect:

1. **Option 1: $ r $**:
   - If the length of the chord were $ r $, it would imply that the chord is equal to the radius, which is not possible in this configuration where the circles intersect such that each passes through the center of the other.

2. **Option 2: $ \sqrt{2}r $**:
   - This length would imply a different geometric relationship, typically associated with the diagonal of a square, not applicable in this scenario of intersecting circles.

3. **Option 3: $ \frac{\sqrt{3}}{2}r $**:
   - This value represents half the length of the chord segment AM, not the full length of the chord AB. Therefore, it cannot be the correct length of the entire chord.

In conclusion, the correct length of the common chord is $\sqrt{3}r$, making Option 4 the correct choice.