Geometry Cheatsheet
Basic Concepts
- Geometry is the study of shapes, sizes, positions, and dimensions of objects in space.
- Euclidean geometry is the study of geometry based on Euclid’s axioms.
- Non-Euclidean geometry is the study of geometry that does not satisfy Euclid’s axioms.
Euclidean Geometry
- Euclidean geometry is based on five axioms, including the parallel postulate, which states that given a line and a point not on the line, there is exactly one line through the point that is parallel to the given line.
- Euclidean geometry includes concepts such as points, lines, planes, angles, circles, and polygons.
- The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Non-Euclidean Geometry
- Non-Euclidean geometry includes hyperbolic geometry and elliptic geometry.
- Hyperbolic geometry satisfies all of Euclid’s axioms except the parallel postulate.
- Elliptic geometry satisfies all of Euclid’s axioms, but the parallel postulate is replaced by its negation.
Coordinate Geometry
- Coordinate geometry is the study of geometry using coordinates.
- The distance formula gives the distance between two points in a coordinate system.
- The midpoint formula gives the coordinates of the midpoint of a line segment.
Trigonometry
- Trigonometry is the study of the relationships between the sides and angles of triangles.
- The sine, cosine, and tangent functions can be used to calculate the lengths of sides and measures of angles in right triangles.
- The law of sines and the law of cosines can be used to solve triangles that are not right triangles.
- Transformations are functions that map points in a plane to other points in the same plane.
- Common transformations include translations, reflections, rotations, and dilations.
- Transformations can be used to study symmetry and congruence.
Resources