1. **State the problem:** We have two concentric circles. The outer circle has a radius of 10 cm. The area of the inner circle is half the area of the outer circle. We need to find the border radius between the two circles, which is the radius of the inner circle. 2. **Formula for the area of a circle:** $$A = \pi r^2$$ where $r$ is the radius. 3. **Calculate the area of the outer circle:** $$A_{outer} = \pi \times 10^2 = 100\pi$$ 4. **Given that the area of the inner circle is half the outer circle:** $$A_{inner} = \frac{1}{2} A_{outer} = \frac{1}{2} \times 100\pi = 50\pi$$ 5. **Find the radius of the inner circle using the area formula:** $$A_{inner} = \pi r_{inner}^2 = 50\pi$$ Divide both sides by $\pi$: $$\cancel{\pi} r_{inner}^2 = 50 \cancel{\pi}$$ $$r_{inner}^2 = 50$$ 6. **Take the square root of both sides:** $$r_{inner} = \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$$ 7. **Final answer:** The border radius between the two circles is $5\sqrt{2}$ cm, approximately 7.07 cm.