Algebra vs Geometry

Variables and Expressions

Variables are symbols, usually letters, that represent unknown or changeable quantities. Algebraic expressions combine variables, constants, and operations such as addition, subtraction, multiplication, and division into meaningful mathematical phrases.

Linear Equations

A linear equation is an equation in which the highest power of the variable is one. These equations graph as straight lines on the coordinate plane and have the general form $ax + b = c$. They are solved by isolating the variable using inverse operations.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, written in the standard form $ax^2 + bx + c = 0$. They can be solved by factoring, completing the square, or using the quadratic formula. Their graphs are parabolas that open upward or downward.

Systems of Equations

A system of equations is a set of two or more equations with the same variables that must be satisfied simultaneously. Common methods for solving systems include substitution, elimination, and graphing. A system can have one solution, no solution, or infinitely many solutions.

Polynomials

Polynomials are expressions consisting of variables raised to non-negative integer powers, multiplied by coefficients, and combined using addition or subtraction. The degree of a polynomial is the highest exponent of its variable. Operations on polynomials include addition, subtraction, multiplication, and division.

Euclidean Geometry

The study of geometry based on Euclid's five postulates, dealing with flat (planar) space and three-dimensional space. It forms the foundation of most high-school geometry, covering congruence, similarity, parallelism, and the properties of common shapes like triangles, circles, and polygons.

Pythagorean Theorem

A fundamental relation in Euclidean geometry stating that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides ($a^2 + b^2 = c^2$). It connects algebra and geometry and serves as the basis for the distance formula in coordinate geometry.

Congruence and Similarity

Two figures are congruent if they have the same shape and size, and similar if they have the same shape but possibly different sizes. Congruence criteria (SSS, SAS, ASA, AAS) and similarity criteria (AA, SAS, SSS) are essential tools for proving geometric relationships.

Coordinate Geometry (Analytic Geometry)

A method of studying geometry by placing figures on a coordinate plane and using algebraic equations to describe them. This approach allows geometric problems to be solved with algebraic techniques such as finding distances, midpoints, and slopes.

Transformations

Operations that move or change geometric figures while preserving certain properties. The four main rigid transformations (isometries) are translation, rotation, reflection, and glide reflection, each preserving distance and angle measure. Dilations change size but preserve shape.