Calculate the length (in cm) of the chord of the circle which is at the distance of the 12 cm from the centre and at the radius of the circle is 13 cm.

Options

A. 10
B. 12
C. 13
D. 15

Answer

To find the length of the chord of the circle that is at a distance of 12 cm from the center, with a radius of 13 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Distance from the center to the chord (perpendicular distance) = 12 cm
- Radius of the circle = 13 cm

2. **Draw a Diagram:**
- Draw a circle with center O.
- Draw a chord AB such that the perpendicular distance from O to AB is 12 cm. Let the point where the perpendicular meets the chord be M (the midpoint of AB).

3. **Apply the Pythagorean Theorem:**
- In the right triangle OMA, where:
- OM = 12 cm (perpendicular distance from the center to the chord)
- OA = radius of the circle = 13 cm
- AM = half the length of the chord (which we need to find).
- According to the Pythagorean theorem:
\[ OA^2 = OM^2 + AM^2 \]

4. **Substitute the Known Values:**
- Substitute the values into the equation:
\[ 13^2 = 12^2 + AM^2 \]
- This simplifies to:
\[ 169 = 144 + AM^2 \]

5. **Solve for AM:**
- Rearranging gives:
\[ AM^2 = 169 - 144 \]
\[ AM^2 = 25 \]
- Taking the square root:
\[ AM = 5 \text{ cm} \]

6. **Calculate the Length of the Chord AB:**
- Since AM is half the length of the chord AB:
\[ AB = 2 \times AM = 2 \times 5 = 10 \text{ cm} \]

### Final Answer:
The length of the chord AB is **10 cm**.