The length of common chord of two intersecting circles is 30 cm. If the diameters of these two circles be 50 cm and 34 cm. Calculate the distance between their centres.

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the radii of the circles
The diameters of the two circles are given as 50 cm and 34 cm. We can find the radii by dividing the diameters by 2.

- Radius of Circle 1 (C1) = Diameter of C1 / 2 = 50 cm / 2 = 25 cm
- Radius of Circle 2 (C2) = Diameter of C2 / 2 = 34 cm / 2 = 17 cm

### Step 2: Find the length of half the common chord
The length of the common chord (PQ) is given as 30 cm. The midpoint of the chord divides it into two equal segments.

- Length of PR (half of PQ) = PQ / 2 = 30 cm / 2 = 15 cm

### Step 3: Use the Pythagorean theorem in triangle ARP
In triangle ARP, where A is the center of Circle 1, R is the midpoint of the chord, and P is a point on the chord:

- AP (the radius of Circle 1) = 25 cm
- PR (half the chord) = 15 cm

Using the Pythagorean theorem:
\[ AP^2 = AR^2 + PR^2 \]
\[ 25^2 = AR^2 + 15^2 \]
\[ 625 = AR^2 + 225 \]
\[ AR^2 = 625 - 225 \]
\[ AR^2 = 400 \]
\[ AR = \sqrt{400} = 20 \text{ cm} \]

### Step 4: Use the Pythagorean theorem in triangle BRP
In triangle BRP, where B is the center of Circle 2:

- BP (the radius of Circle 2) = 17 cm
- PR (half the chord) = 15 cm

Using the Pythagorean theorem:
\[ BP^2 = RB^2 + PR^2 \]
\[ 17^2 = RB^2 + 15^2 \]
\[ 289 = RB^2 + 225 \]
\[ RB^2 = 289 - 225 \]
\[ RB^2 = 64 \]
\[ RB = \sqrt{64} = 8 \text{ cm} \]

### Step 5: Calculate the distance between the centers of the circles
The distance between the centers A and B (AB) is the sum of AR and RB:

\[ AB = AR + RB \]
\[ AB = 20 \text{ cm} + 8 \text{ cm} = 28 \text{ cm} \]

### Final Answer
The distance between the centers of the two circles is **28 cm**.

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