Co-ordinate Geometry

Definition: Branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids.
Basic Concepts:
- Point: An exact location in space with no size or dimension.
- Line: A straight one-dimensional figure having no thickness and extending infinitely in both directions.
- Plane: A flat, two-dimensional surface that extends infinitely in all directions.
Shapes and Figures:
- Triangles: Three-sided polygons classified by angles (acute, right, obtuse) and side lengths (equilateral, isosceles, scalene).
- Quadrilaterals: Four-sided figures (e.g., squares, rectangles, trapezoids).
- Circles: A set of points equidistant from a central point, defined by radius and diameter.
Properties:
- Angles: Formed by two rays; types include acute, right, obtuse, and straight.
- Congruence: Two shapes are congruent if they have the same size and shape.
- Similarity: Two shapes are similar if they have the same shape but different sizes.
Theorems and Formulas:
- Pythagorean Theorem: In a right triangle, (a^2 + b^2 = c^2), where (c) is the hypotenuse.
- Area Formulas:
  - Rectangle: (A = l \times w)
  - Triangle: (A = \frac{1}{2} \times b \times h)
  - Circle: (A = \pi r^2)
- Volume Formulas:
  - Cylinder: (V = \pi r^2 h)
  - Sphere: (V = \frac{4}{3} \pi r^3)

Definition: The study of geometric figures using a coordinate system, typically the Cartesian plane.
Coordinate System:
- Axes: The x-axis (horizontal) and y-axis (vertical) intersect at the origin (0,0).
- Coordinates: Each point in the plane is represented by an ordered pair ((x, y)).
Distance Formula:
- The distance (d) between two points ((x_1, y_1)) and ((x_2, y_2)) is given by: [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]
Midpoint Formula:
- The midpoint (M) of a line segment between ((x_1, y_1)) and ((x_2, y_2)) is: [ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) ]
Slope:
- The slope (m) of a line through points ((x_1, y_1)) and ((x_2, y_2)) is calculated as: [ m = \frac{y_2 - y_1}{x_2 - x_1} ]
Equation of a Line:
- Slope-Intercept Form: (y = mx + b), where (m) is the slope and (b) is the y-intercept.
- Point-Slope Form: (y - y_1 = m(x - x_1)), using a point ((x_1, y_1)) on the line.
Conic Sections:
- Circle: Defined by ((x - h)^2 + (y - k)^2 = r^2).
- Ellipse: (\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1).
- Parabola: (y = ax^2 + bx + c) or (x = ay^2 + by + c).
- Hyperbola: (\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1).

Geometry studies properties and relationships of figures, including points, lines, surfaces, and solids.
A point represents an exact location in space without size or dimension.
A line is a straight, one-dimensional figure that extends infinitely without thickness.
A plane is a flat, two-dimensional surface that extends infinitely in all directions.
Triangles, three-sided polygons, are classified by angles (acute, right, obtuse) and sides (equilateral, isosceles, scalene).
Quadrilaterals are four-sided shapes, such as squares, rectangles, and trapezoids.
A circle is defined as a set of points equidistant from a central point, characterized by radius and diameter.
Angles are formed by two rays and categorized into types: acute, right, obtuse, and straight.
Congruence indicates two shapes with the same size and shape, while similarity denotes shapes with the same shape but different sizes.
The Pythagorean Theorem states that in a right triangle, (a^2 + b^2 = c^2) (where (c) is the hypotenuse).
Area of a rectangle is calculated as (A = l \times w); for a triangle, (A = \frac{1}{2} \times b \times h); for a circle, (A = \pi r^2).
Volume for a cylinder is (V = \pi r^2 h) and for a sphere, (V = \frac{4}{3} \pi r^3).

The study of geometric figures within a coordinate system, mostly the Cartesian plane.
The coordinate system consists of two axes: the x-axis (horizontal) and y-axis (vertical), intersecting at the origin (0,0).
Coordinates of points in the plane are expressed as ordered pairs ((x, y)).
The distance formula computes the distance (d) between two points ((x_1, y_1)) and ((x_2, y_2)) as: [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]
The midpoint of a line segment between ((x_1, y_1)) and ((x_2, y_2)) is given by:
[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) ]
Slope (m) of a line through two points can be found using: [ m = \frac{y_2 - y_1}{x_2 - x_1} ]
The slope-intercept form of a line is expressed as (y = mx + b) with (m) as the slope and (b) as the y-intercept.
The point-slope form is articulated as (y - y_1 = m(x - x_1)) using a specific point ((x_1, y_1)) on the line.
Conic sections include:
- Circle: defined as ((x - h)^2 + (y - k)^2 = r^2)
- Ellipse: described by (\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1)
- Parabola: represented as (y = ax^2 + bx + c) or (x = ay^2 + by + c)
- Hyperbola: expressed as (\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1)