A chord of length 7 cm subtends an angle of `60^@` at the centre of a circle. What is the radius (in cm) of the circle?

Options

A. `7sqrt(2)`
B. `7sqrt(3)`
C. `7`
D. `14`

Answer

To find the radius of the circle given that a chord of length 7 cm subtends an angle of 60 degrees at the center, we can follow these steps:

### Step 1: Understand the Geometry
We have a circle with center O, and a chord AB that is 7 cm long. The angle ∠AOB at the center O is 60 degrees.

### Step 2: Draw the Triangle
Draw triangle OAB where:
- OA and OB are the radii of the circle (both equal to r).
- AB is the chord of length 7 cm.

### Step 3: Apply the Law of Cosines
In triangle OAB, we can use the Law of Cosines to relate the sides and the angle:
\[ AB^2 = OA^2 + OB^2 - 2 \cdot OA \cdot OB \cdot \cos(\angle AOB) \]
Since OA = OB = r (the radius), we can rewrite it as:
\[ AB^2 = r^2 + r^2 - 2r^2 \cos(60^\circ) \]

### Step 4: Substitute Known Values
We know that:
- AB = 7 cm
- cos(60°) = 0.5

Substituting these values into the equation:
\[ 7^2 = r^2 + r^2 - 2r^2 \cdot 0.5 \]
\[ 49 = 2r^2 - r^2 \]
\[ 49 = r^2 \]

### Step 5: Solve for r
Now, we can solve for r:
\[ r^2 = 49 \]
\[ r = \sqrt{49} \]
\[ r = 7 \text{ cm} \]

### Conclusion
The radius of the circle is 7 cm.

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