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 that lie on a flat surface (a plane). - **Key concepts:** Points, lines, angles, triangles, quadrilaterals, circles. - **Important formulas:** - Triangle area: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ - Circle circumference: $$C = 2\pi r$$ - Circle area: $$A = \pi r^2$$ - **Rules and theorems:** - Sum of angles in a triangle is $$180^\circ$$. - Pythagoras theorem: In a right triangle, $$a^2 + b^2 = c^2$$ where $$c$$ is the hypotenuse. 3. **Solid Geometry:** Solid geometry studies three-dimensional shapes like cubes, spheres, cylinders, and cones. - **Key concepts:** Volume, surface area, edges, faces, vertices. - **Important 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 of sphere: $$A = 4 \pi r^2$$ - **Rules and theorems:** - Euler’s formula for polyhedra: $$V - E + F = 2$$ where $$V$$ is vertices, $$E$$ edges, $$F$$ faces. 4. **Coordinate Geometry:** Coordinate geometry uses algebra and coordinates to study geometric figures. - **Key concepts:** Points represented as $$(x,y)$$ in 2D or $$(x,y,z)$$ in 3D. - **Important formulas:** - Distance between two points: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ - Midpoint of segment: $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ - Equation of a line: $$y = mx + c$$ where $$m$$ is slope, $$c$$ is y-intercept. - **Rules and theorems:** - Slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ - Two lines are perpendicular if $$m_1 \times m_2 = -1$$. This overview covers the theory, formulas, and key rules/theorems of plane, solid, and coordinate geometry.