Circle of radius 10 cm and 8 cm intersects at A and B AB = 12 find the distance b/w centres

Answer

To find the distance between the centers of two intersecting circles with given radii and the length of the chord, we can follow these steps:

### Step 1: Identify the given values
- Radius of Circle 1 (R1) = 10 cm
- Radius of Circle 2 (R2) = 8 cm
- Length of chord AB = 12 cm

### Step 2: Find the half-length of the chord
Since the chord AB is divided into two equal parts by the perpendicular from the center to the chord:
- Length of AO = OB = AB/2 = 12 cm / 2 = 6 cm

### Step 3: Apply the Pythagorean theorem for Circle 1
Let C1 be the center of Circle 1. We can use the Pythagorean theorem in triangle AOC1:
- C1O^2 + AO^2 = AC1^2
- C1O^2 + 6^2 = 10^2
- C1O^2 + 36 = 100
- C1O^2 = 100 - 36
- C1O^2 = 64
- C1O = √64 = 8 cm

### Step 4: Apply the Pythagorean theorem for Circle 2
Let C2 be the center of Circle 2. We can use the Pythagorean theorem in triangle AOC2:
- C2O^2 + AO^2 = AC2^2
- C2O^2 + 6^2 = 8^2
- C2O^2 + 36 = 64
- C2O^2 = 64 - 36
- C2O^2 = 28
- C2O = √28 = 2√7 cm

### Step 5: Calculate the distance between the centers
The distance between the centers C1 and C2 is given by:
- C1C2 = C1O + C2O
- C1C2 = 8 cm + 2√7 cm

### Final Answer
Thus, the distance between the centers of the two circles is:
- C1C2 = 8 + 2√7 cm

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