A chord of length 24 cm is at a distance of 5 cm form the centre of the circle. Find the length of another chord of circle which is at a distance of 12 cm from the centre.

Explanation

To solve the problem step by step, we will use the properties of circles and the Pythagorean theorem.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Length of the first chord (AB) = 24 cm
- Distance from the center (O) to the first chord = 5 cm

2. **Divide the Chord into Two Equal Parts:**
- Since the perpendicular from the center to the chord bisects the chord, each half of the chord (AM and MB) will be:
\[ AM = MB = \frac{24}{2} = 12 \text{ cm} \]

3. **Apply the Pythagorean Theorem:**
- In triangle OMA (where O is the center, M is the midpoint of the chord, and A is one endpoint of the chord):
- OM = 5 cm (distance from the center to the chord)
- AM = 12 cm (half the length of the chord)
- OA = radius of the circle (which we need to find)
- According to Pythagorean theorem:
\[ OA^2 = OM^2 + AM^2 \]
\[ OA^2 = 5^2 + 12^2 \]
\[ OA^2 = 25 + 144 = 169 \]
\[ OA = \sqrt{169} = 13 \text{ cm} \]

4. **Find the Length of the Second Chord:**
- Now, we need to find the length of another chord (CD) that is at a distance of 12 cm from the center.
- Let the distance from the center O to the second chord be OD = 12 cm.
- We will again use the Pythagorean theorem in triangle OCD (where C is the midpoint of the second chord):
- OC = radius of the circle = 13 cm
- OD = 12 cm (distance from the center to the chord)
- Let the half-length of the chord be x (i.e., CD = 2x).
- According to Pythagorean theorem:
\[ OC^2 = OD^2 + CD^2 \]
\[ 13^2 = 12^2 + x^2 \]
\[ 169 = 144 + x^2 \]
\[ x^2 = 169 - 144 = 25 \]
\[ x = \sqrt{25} = 5 \text{ cm} \]
- Therefore, the total length of chord CD is:
\[ CD = 2x = 2 \times 5 = 10 \text{ cm} \]

5. **Final Answer:**
- The length of the second chord is **10 cm**.