Two circles of equal radii intersect each other such that one circle will pass through the centre of the other circle. The distance between the centre of two circles is equal to 6 cm. Find the length (in cm) of common chord.

Options

A. `sqrt3`
B. `2sqrt(3)`
C. `3sqrt(3)`
D. `6sqrt(3)`

Answer

To solve the problem of finding the length of the common chord between two intersecting circles of equal radii, we can follow these steps:

### Step 1: Understand the Geometry
Let the radius of each circle be \( r \). The distance between the centers of the two circles is given as \( d = 6 \) cm. Since one circle passes through the center of the other, we know that the radius \( r \) must be equal to \( d \), which means \( r = 6 \) cm.

### Step 2: Find the Length of the Common Chord
The common chord of two intersecting circles can be found using the formula:
\[ L = 2 \sqrt{r^2 - \left(\frac{d}{2}\right)^2} \]
where \( L \) is the length of the common chord, \( r \) is the radius of the circles, and \( d \) is the distance between the centers of the circles.

### Step 3: Substitute the Values
Now, substituting \( r = 6 \) cm and \( d = 6 \) cm into the formula:
\[ L = 2 \sqrt{6^2 - \left(\frac{6}{2}\right)^2} \]
Calculating the values:
\[ L = 2 \sqrt{36 - 3^2} \]
\[ L = 2 \sqrt{36 - 9} \]
\[ L = 2 \sqrt{27} \]
\[ L = 2 \times 3\sqrt{3} \]
\[ L = 6\sqrt{3} \text{ cm} \]

### Step 4: Conclusion
Thus, the length of the common chord is \( 6\sqrt{3} \) cm.

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