1. **State the problem:** We have a circle with center A, and an arc BC whose length is \(\frac{2}{5}\) of the entire circumference. We need to find the central angle \(x^\circ\) subtended by arc BC. 2. **Recall the formula for arc length:** The length of an arc \(s\) is related to the radius \(r\) and the central angle \(\theta\) in radians by $$s = r\theta$$ 3. **Recall the circumference formula:** The circumference \(C\) of a circle is $$C = 2\pi r$$ 4. **Given:** $$s = \frac{2}{5}C = \frac{2}{5} \times 2\pi r = \frac{4\pi r}{5}$$ 5. **Use the arc length formula:** $$s = r\theta \implies \frac{4\pi r}{5} = r\theta$$ Divide both sides by \(r\) (assuming \(r \neq 0\)): $$\theta = \frac{4\pi}{5}$$ 6. **Convert \(\theta\) from radians to degrees:** $$x = \theta \times \frac{180}{\pi} = \frac{4\pi}{5} \times \frac{180}{\pi} = \frac{4 \times 180}{5} = 144^\circ$$ **Final answer:** \(x = 144^\circ\)