1. **Problem statement:** We want to find how a line divides a sector into two equal areas. 2. **Understanding the sector:** A sector is a portion of a circle bounded by two radii and the arc between them. 3. **Formula for sector area:** The area of a sector with radius $r$ and central angle $\theta$ (in radians) is given by $$\text{Area} = \frac{1}{2} r^2 \theta$$ 4. **Dividing the sector:** To divide the sector into two equal areas, the line must create two smaller sectors each with half the original area. 5. **Finding the dividing line:** If the original sector has angle $\theta$, the line dividing it into two equal areas will create two sectors with angles $\alpha$ and $\theta - \alpha$ such that $$\frac{1}{2} r^2 \alpha = \frac{1}{2} r^2 (\theta - \alpha) = \frac{1}{2} \times \text{original area}$$ 6. **Solving for $\alpha$:** Since the areas are equal, $$\alpha = \theta - \alpha$$ $$2\alpha = \theta$$ $$\alpha = \frac{\theta}{2}$$ 7. **Conclusion:** The line that divides the sector into two equal areas is the radius that bisects the central angle $\theta$ into two equal angles of $\frac{\theta}{2}$ each. This means the dividing line is the angle bisector of the sector's central angle.