Two equal circles of radius 4 cm intersect each other such that each passes through the centre of the other. The length of the common chord is :

Options

A. `2sqrt(3)` cm
B. `4sqrt(3)` cm
C. `2sqrt(2)` cm
D. 8 cm

Answer

To find the length of the common chord of two equal circles that intersect such that each passes through the center of the other, we can follow these steps:

### Step 1: Understand the Geometry
We have two circles, each with a radius of 4 cm. Since each circle passes through the center of the other, the distance between the centers of the circles is equal to the radius, which is 4 cm.

### Step 2: Draw the Diagram
Draw two circles with centers A and B. Mark the radius of each circle as 4 cm. The points where the circles intersect will be labeled as P and Q, forming the common chord PQ.

### Step 3: Identify the Triangle Formed
The centers A and B, along with the intersection points P and Q, form an isosceles triangle (triangle ABP or triangle ABQ). The lengths of AP and BP are both equal to the radius of the circles, which is 4 cm.

### Step 4: Calculate the Length of the Common Chord
To find the length of the common chord PQ, we can use the property of the isosceles triangle. The height from the midpoint of PQ to the line AB will bisect the chord PQ and will also be perpendicular to AB.

1. Let M be the midpoint of PQ.
2. The length AM (the height) can be calculated using the Pythagorean theorem in triangle AMP:
- AM = √(AP² - PM²)
- Since PM = PQ/2, we need to find PM.

### Step 5: Use the Pythagorean Theorem
In triangle ABM:
- AB = 4 cm (the distance between the centers)
- AM = height from A to line PQ
- BM = height from B to line PQ

Since AM is the same as BM, we can find the length of PM:
- Let PM = x, then PQ = 2x.

Using the Pythagorean theorem:
- AM² + PM² = AP²
- AM² + x² = 4²
- AM² + x² = 16

### Step 6: Find the Length of the Common Chord
Since the triangles are symmetric, we can find the height AM:
- The distance AB is equal to the radius, which is 4 cm. Therefore, the height AM can be calculated as:
- AM = √(4² - (PQ/2)²)
- Since AM = 2√3 (as derived from the equilateral triangle properties), we can substitute this back into the equation to find PQ.

### Step 7: Final Calculation
Using the derived height:
- 2√3 = √(16 - x²)
- Squaring both sides gives us:
- 12 = 16 - x²
- x² = 4
- x = 2

Thus, the length of the common chord PQ is:
- PQ = 2 * PM = 2 * 2 = 4 cm.

### Final Answer
The length of the common chord PQ is **4 cm**.