1. **Stating the problem:** You asked for an explanation of plane geometry, solid geometry, and coordinate geometry including theory, formulas, rules, and theorems. 2. **Plane Geometry:** - Plane geometry studies shapes like points, lines, circles, and polygons on a flat surface (2D). - Important concepts: points (location), lines (straight paths), angles (formed by two lines), triangles, quadrilaterals. - Key formulas: - Triangle area: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ - Circle circumference: $$C = 2\pi r$$ - Circle area: $$A = \pi r^2$$ - Important theorems: - Pythagoras theorem: $$a^2 + b^2 = c^2$$ for right triangles. - Triangle sum theorem: sum of interior angles = $$180^\circ$$. 3. **Solid Geometry:** - Solid geometry studies 3D shapes like cubes, spheres, cylinders, cones. - Key formulas: - Cube volume: $$V = s^3$$ where $$s$$ is side length. - Sphere volume: $$V = \frac{4}{3} \pi r^3$$ - Cylinder volume: $$V = \pi r^2 h$$ - Surface area formulas also important. 4. **Coordinate Geometry:** - Combines algebra and geometry using coordinates on a plane. - Points represented as $$(x,y)$$. - Distance formula between points $$(x_1,y_1)$$ and $$(x_2,y_2)$$: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ - Midpoint formula: $$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$ - Equation of a line: $$y = mx + c$$ where $$m$$ is slope and $$c$$ is y-intercept. - Slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 5. **Summary:** - Plane geometry deals with flat shapes and their properties. - Solid geometry studies 3D objects and their volumes and surface areas. - Coordinate geometry uses algebraic methods to study geometric problems on coordinate planes. This covers the basic theory, formulas, rules, and key theorems for each type of geometry.