The length of the radius of a circle is one-half the length of an arc of the circle . What is the radian measure of the central angle that intercepts that arc ?

Options

A. `60^(@)`
B. `120^(@)`
C. `1^(R)`
D. `2^(R)`

Explanation

To solve the problem, we need to find the radian measure of the central angle that intercepts an arc, given that the length of the radius of the circle is one-half the length of the arc.

### Step-by-Step Solution:

1. **Define Variables**:
Let \( R \) be the radius of the circle and \( L \) be the length of the arc. According to the problem, we have:
\[ R = \frac{1}{2} L \]

2. **Use the Arc Length Formula**:
The formula for the length of an arc in terms of the radius and the central angle in radians is:
\[ L = R \theta \]
where \( \theta \) is the central angle in radians.

3. **Substitute for \( L \)**:
From the first step, we can express \( L \) in terms of \( R \):
\[ L = 2R \]
Now, substitute this expression for \( L \) into the arc length formula:
\[ 2R = R \theta \]

4. **Simplify the Equation**:
We can divide both sides of the equation by \( R \) (assuming \( R \neq 0 \)):
\[ 2 = \theta \]

5. **Conclusion**:
Therefore, the radian measure of the central angle \( \theta \) that intercepts the arc is:
\[ \theta = 2 \text{ radians} \]

### Final Answer:
The radian measure of the central angle that intercepts the arc is \( \theta = 2 \) radians.