In a circle 10 cm long chord is at a distance of 12 cm form the centre. Find the length of a chord at a distance of 5cm from the centre.

Explanation

To solve the problem step by step, we will follow the geometric principles related to circles and chords.

### Step 1: Understand the given information
We have a chord AB of length 10 cm, which is at a distance of 12 cm from the center O of the circle. We need to find the length of another chord XY that is at a distance of 5 cm from the center O.

### Step 2: Find the radius of the circle
1. Since the perpendicular from the center O to the chord AB bisects the chord, we can denote the midpoint of AB as P.
2. Therefore, AP = PB = 10 cm / 2 = 5 cm.
3. We can form a right triangle OAP, where:
- OP = 12 cm (distance from the center to the chord)
- AP = 5 cm (half the length of the chord)

Using the Pythagorean theorem:
\[ OA^2 = OP^2 + AP^2 \]
\[ OA^2 = 12^2 + 5^2 \]
\[ OA^2 = 144 + 25 \]
\[ OA^2 = 169 \]
\[ OA = \sqrt{169} = 13 \text{ cm} \]

### Step 3: Find the length of the chord at a distance of 5 cm from the center
1. Let the chord XY be at a distance of 5 cm from the center O. We can denote the midpoint of XY as M.
2. We can form another right triangle OXM, where:
- OM = 5 cm (distance from the center to the chord)
- OX = OA = 13 cm (radius of the circle)

Using the Pythagorean theorem again:
\[ OX^2 = OM^2 + MX^2 \]
\[ 13^2 = 5^2 + MX^2 \]
\[ 169 = 25 + MX^2 \]
\[ MX^2 = 169 - 25 \]
\[ MX^2 = 144 \]
\[ MX = \sqrt{144} = 12 \text{ cm} \]

### Step 4: Calculate the length of chord XY
Since M is the midpoint of XY, we have:
\[ XY = 2 \times MX = 2 \times 12 = 24 \text{ cm} \]

### Conclusion
The length of the chord XY at a distance of 5 cm from the center is **24 cm**.

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