The length of the common chord of two intersecting circles is 24 cm. If the diameter of the circles are 30 cm and 26 cm, then the distance between the centre (in cm) is

Options

A. 13
B. 14
C. 15
D. 16

Answer

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

### Step 1: Identify the radii of the circles
The diameters of the circles are given as:
- Circle 1: Diameter = 30 cm, so Radius \( r_1 = \frac{30}{2} = 15 \) cm
- Circle 2: Diameter = 26 cm, so Radius \( r_2 = \frac{26}{2} = 13 \) cm

### Step 2: Determine the length of the common chord
The length of the common chord \( AB \) is given as 24 cm. Therefore, the half-length of the chord \( AP \) is:
\[ AP = \frac{AB}{2} = \frac{24}{2} = 12 \text{ cm} \]

### Step 3: Use the Pythagorean theorem in triangle \( O_1AP \)
In triangle \( O_1AP \):
- \( O_1A \) is the radius of Circle 1, which is \( r_1 = 15 \) cm.
- \( AP = 12 \) cm.
- Let \( O_1P \) be the distance from the center of Circle 1 to the midpoint of the chord.

Using the Pythagorean theorem:
\[ O_1A^2 = AP^2 + O_1P^2 \]
Substituting the values:
\[ 15^2 = 12^2 + O_1P^2 \]
\[ 225 = 144 + O_1P^2 \]
\[ O_1P^2 = 225 - 144 = 81 \]
\[ O_1P = \sqrt{81} = 9 \text{ cm} \]

### Step 4: Use the Pythagorean theorem in triangle \( O_2AP \)
In triangle \( O_2AP \):
- \( O_2A \) is the radius of Circle 2, which is \( r_2 = 13 \) cm.
- \( AP = 12 \) cm.
- Let \( O_2P \) be the distance from the center of Circle 2 to the midpoint of the chord.

Using the Pythagorean theorem:
\[ O_2A^2 = AP^2 + O_2P^2 \]
Substituting the values:
\[ 13^2 = 12^2 + O_2P^2 \]
\[ 169 = 144 + O_2P^2 \]
\[ O_2P^2 = 169 - 144 = 25 \]
\[ O_2P = \sqrt{25} = 5 \text{ cm} \]

### Step 5: Find the distance between the centers \( O_1O_2 \)
The distance between the centers of the circles can be calculated by adding the distances \( O_1P \) and \( O_2P \):
\[ O_1O_2 = O_1P + O_2P = 9 + 5 = 14 \text{ cm} \]

### Final Answer
The distance between the centers of the two circles is **14 cm**.

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