1. **Problem Statement:** We have a circle with two points A and B on its circumference. Line AB is not a diameter. We need to order the following parts of the circle by their areas in ascending order: - Major sector formed by two radii and major arc AB - Minor sector formed by two radii and minor arc AB - Major segment formed by line AB and major arc AB - Minor segment formed by line AB and minor arc AB 2. **Key Definitions and Formulas:** - A **sector** is the region bounded by two radii and the arc between them. - A **segment** is the region bounded by a chord (line AB) and the arc it subtends. - The **minor arc** is the smaller arc between points A and B; the **major arc** is the larger one. Formulas: - Area of sector with central angle $\theta$ (in radians): $$\text{Area}_{sector} = \frac{1}{2} r^2 \theta$$ - Area of segment with central angle $\theta$: $$\text{Area}_{segment} = \frac{1}{2} r^2 (\theta - \sin \theta)$$ 3. **Important Notes:** - Since AB is not a diameter, $0 < \theta < 2\pi$ and $\theta \neq \pi$. - The minor sector corresponds to angle $\theta$. - The major sector corresponds to angle $2\pi - \theta$. - Similarly for segments. 4. **Comparing Areas:** - Minor sector area: $\frac{1}{2} r^2 \theta$ - Major sector area: $\frac{1}{2} r^2 (2\pi - \theta)$ - Minor segment area: $\frac{1}{2} r^2 (\theta - \sin \theta)$ - Major segment area: $\frac{1}{2} r^2 ((2\pi - \theta) - \sin(2\pi - \theta))$ Since $\sin(2\pi - \theta) = -\sin \theta$, the major segment area becomes: $$\frac{1}{2} r^2 ((2\pi - \theta) + \sin \theta)$$ 5. **Ordering:** - Minor segment area $= \frac{1}{2} r^2 (\theta - \sin \theta)$ - Minor sector area $= \frac{1}{2} r^2 \theta$ - Major segment area $= \frac{1}{2} r^2 ((2\pi - \theta) + \sin \theta)$ - Major sector area $= \frac{1}{2} r^2 (2\pi - \theta)$ Since $0 < \theta < \pi$, we have $\sin \theta > 0$. Comparisons: - $\theta - \sin \theta < \theta$ so minor segment area < minor sector area. - $(2\pi - \theta) < (2\pi - \theta) + \sin \theta$ so major sector area < major segment area. Also, minor sector area $< $ major sector area because $\theta < 2\pi - \theta$. 6. **Final ascending order of areas:** $$\text{minor segment} < \text{minor sector} < \text{major sector} < \text{major segment}$$