A chord of length 120 cm is at a distance of 32 cm from the centre of a circle. What is the radius (in cm) of the circle?

Options

A. 96
B. 34
C. 72
D. 68

Answer

To find the radius of the circle given a chord of length 120 cm that is 32 cm away from the center of the circle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components**:
- Let O be the center of the circle.
- Let AB be the chord of length 120 cm.
- Let M be the midpoint of the chord AB.
- The distance from the center O to the chord AB is given as OM = 32 cm.

2. **Calculate Half the Length of the Chord**:
- Since M is the midpoint of the chord AB, we can find AM (half the length of the chord).
- Length of chord AB = 120 cm, so AM = AB / 2 = 120 cm / 2 = 60 cm.

3. **Use the Pythagorean Theorem**:
- In the right triangle OAM, we can apply the Pythagorean theorem:
\[ OA^2 = OM^2 + AM^2 \]
- Here, OA is the radius (which we need to find), OM is the distance from the center to the chord (32 cm), and AM is half the chord length (60 cm).

4. **Substitute the Values**:
- Substitute the known values into the equation:
\[ OA^2 = 32^2 + 60^2 \]

5. **Calculate the Squares**:
- Calculate \(32^2\) and \(60^2\):
\[ 32^2 = 1024 \]
\[ 60^2 = 3600 \]

6. **Add the Squares**:
- Now add these two results:
\[ OA^2 = 1024 + 3600 = 4624 \]

7. **Find the Radius**:
- To find OA (the radius), take the square root of 4624:
\[ OA = \sqrt{4624} = 68 \text{ cm} \]

### Final Answer:
The radius of the circle is **68 cm**.