In a circle with center O, the measure of central angle POQ is `(3pi)/(2)` radians. The length of the arc formed by central angle POQ is that fraction of the circumference of the circle?

Options

A. `(3)/(16)`
B. `(3)/(8)`
C. `(3)/(4)`
D. `(3)/(2)`

Answer

To solve the problem, we need to find the fraction of the circumference of the circle that corresponds to the length of the arc formed by the central angle POQ, which measures \( \frac{3\pi}{2} \) radians.

### Step-by-Step Solution:

1. **Identify the Circumference of the Circle**:
The formula for the circumference \( C \) of a circle with radius \( r \) is given by:
\[ C = 2\pi r \]

2. **Determine the Length of the Arc**:
The length \( L \) of the arc corresponding to a central angle \( \theta \) (in radians) can be calculated using the formula:
\[ L = r \theta \]
Here, the central angle \( \theta \) is given as \( \frac{3\pi}{2} \). Therefore, the length of the arc \( L \) becomes:
\[ L = r \cdot \frac{3\pi}{2} = \frac{3\pi r}{2} \]

3. **Calculate the Fraction of the Circumference**:
To find the fraction of the circumference that the arc length represents, we take the ratio of the arc length \( L \) to the circumference \( C \):
\[ \text{Fraction} = \frac{L}{C} = \frac{\frac{3\pi r}{2}}{2\pi r} \]

4. **Simplify the Fraction**:
In the fraction, we can cancel \( \pi r \) from the numerator and the denominator:
\[ \text{Fraction} = \frac{\frac{3}{2}}{2} = \frac{3}{2} \cdot \frac{1}{2} = \frac{3}{4} \]

5. **Conclusion**:
Thus, the length of the arc formed by the central angle POQ is \( \frac{3}{4} \) of the circumference of the circle.

### Final Answer:
The length of the arc formed by central angle POQ is \( \frac{3}{4} \) of the circumference of the circle.

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