Two circles with same radius r intersect each other and one passes through the centre of the other. Then the length of the common chord is

Options

A. r
B. `sqrt3`r
C. `(sqrt3)/(2)`r
D. `sqrt5`r

Answer

To find the length of the common chord of two intersecting circles of the same radius \( r \), where one circle passes through the center of the other, we can follow these steps:

### Step 1: Understand the Geometry
We have two circles, both with radius \( r \). Let’s denote the centers of the circles as \( O_1 \) and \( O_2 \). The distance between the centers \( O_1 \) and \( O_2 \) is equal to the radius \( r \) since one circle passes through the center of the other.

### Step 2: Identify the Common Chord
The common chord \( AB \) of the two circles can be found by drawing a perpendicular line from the midpoint \( M \) of the chord to the line segment \( O_1O_2 \). This midpoint \( M \) is also the point where the line connecting the centers intersects the common chord.

### Step 3: Use the Right Triangle
In triangle \( O_1MA \):
- \( O_1A \) is the radius \( r \).
- \( O_1M \) is the distance from the center \( O_1 \) to the midpoint \( M \) of the chord.
- \( AM \) is half the length of the common chord, which we will denote as \( x \).

Using the Pythagorean theorem:
\[ O_1A^2 = O_1M^2 + AM^2 \]
Substituting the known values:
\[ r^2 = O_1M^2 + x^2 \]

### Step 4: Find \( O_1M \)
The distance \( O_1M \) can be calculated using the fact that \( O_1O_2 = r \). The distance from the center of one circle to the midpoint of the chord can be derived from the right triangle formed by the centers and the midpoint:
\[ O_1M = \sqrt{O_1O_2^2 - AM^2} \]
Since \( O_1O_2 = r \), we have:
\[ O_1M = \sqrt{r^2 - x^2} \]

### Step 5: Substitute Back into the Equation
Now substituting \( O_1M \) back into the Pythagorean theorem gives:
\[ r^2 = (r^2 - x^2) + x^2 \]
This simplifies to:
\[ r^2 = r^2 - x^2 + x^2 \]
Thus, we can find \( x \) using the relationship:
\[ O_1M = \sqrt{r^2 - x^2} \]

### Step 6: Solve for the Length of the Common Chord
Since the distance \( O_1O_2 = r \) and \( O_1M \) is half the length of the common chord, we can express the length of the common chord \( AB \) as:
\[ AB = 2x = 2 \sqrt{r^2 - \left(\frac{r}{2}\right)^2} \]
Calculating this gives:
\[ AB = 2 \sqrt{r^2 - \frac{r^2}{4}} = 2 \sqrt{\frac{3r^2}{4}} = 2 \cdot \frac{r\sqrt{3}}{2} = r\sqrt{3} \]

### Final Answer
Thus, the length of the common chord is:
\[ \boxed{r\sqrt{3}} \]