The angle between two legs of a compass is `60^(@)` and length of each is 10 cm the distance between end points of the leg is

Options

A. 5 cm
B. 10 cm
C. `5sqrt(3)` cm
D. `10sqrt(3)` cm

Answer

To solve the problem, we need to find the distance between the endpoints of the two legs of a compass given that the angle between them is \(60^\circ\) and the length of each leg is \(10 \, \text{cm}\).

### Step-by-Step Solution:

1. **Identify the Triangle**:
- The two legs of the compass form a triangle with the angle between them as \(60^\circ\). Let's label the points:
- Let point A be the point where the legs meet.
- Let point B be the endpoint of one leg.
- Let point C be the endpoint of the other leg.
- Therefore, we have triangle ABC, where:
- \(AB = 10 \, \text{cm}\) (length of one leg)
- \(AC = 10 \, \text{cm}\) (length of the other leg)
- \(\angle BAC = 60^\circ\).

2. **Use the Cosine Rule**:
- To find the distance \(BC\) (the distance between the endpoints of the legs), we can use the Cosine Rule:
\[ BC^2 = AB^2 + AC^2 - 2 \cdot AB \cdot AC \cdot \cos(\angle BAC) \]
- Substituting the known values:
\[ BC^2 = 10^2 + 10^2 - 2 \cdot 10 \cdot 10 \cdot \cos(60^\circ) \]

3. **Calculate \(\cos(60^\circ)\)**:
- We know that \(\cos(60^\circ) = \frac{1}{2}\).
- Now substituting this value into the equation:
\[ BC^2 = 100 + 100 - 2 \cdot 10 \cdot 10 \cdot \frac{1}{2} \]
\[ BC^2 = 100 + 100 - 100 \]
\[ BC^2 = 100 \]

4. **Find \(BC\)**:
- Taking the square root of both sides:
\[ BC = \sqrt{100} = 10 \, \text{cm} \]

5. **Conclusion**:
- The distance between the endpoints of the legs is \(10 \, \text{cm}\).

### Final Answer:
The distance between the endpoints of the legs is \(10 \, \text{cm}\).