Point X and Y lie on a circle with center C such that the measure of the minor are formed by central angle XCY is `(3)/(10)` of the circumference of the circle. What is the measure of angle XYC, in radians?

Options

A. `(3)/(10)pi`
B. `(3)/(5)pi`
C. `(6)/(5)pi`
D. `(5)/(3)pi`

Answer

To find the measure of angle XYC in radians, we can follow these steps:

### Step 1: Understand the relationship between the arc length and the central angle
The arc length (s) of a circle is related to the radius (r) and the central angle (θ in radians) by the formula:
\[ s = r \cdot \theta \]

### Step 2: Determine the arc length based on the given information
We know that the measure of the minor arc XY is \(\frac{3}{10}\) of the circumference of the circle. The circumference (C) of a circle is given by:
\[ C = 2\pi r \]
Thus, the arc length (s) can be expressed as:
\[ s = \frac{3}{10} \cdot C = \frac{3}{10} \cdot 2\pi r \]

### Step 3: Substitute the arc length into the formula for arc length
Now, substituting the expression for arc length into the formula \(s = r \cdot \theta\):
\[ \frac{3}{10} \cdot 2\pi r = r \cdot \theta \]

### Step 4: Simplify the equation
We can simplify this equation by dividing both sides by \(r\) (assuming \(r \neq 0\)):
\[ \frac{3}{10} \cdot 2\pi = \theta \]

### Step 5: Calculate the value of θ
Now, we can calculate θ:
\[ \theta = \frac{3}{10} \cdot 2\pi = \frac{6\pi}{10} = \frac{3\pi}{5} \]

### Conclusion
Thus, the measure of angle XYC in radians is:
\[ \theta = \frac{3\pi}{5} \]