Introduction to Geometry and Basic Shapes

• Mathematics encompasses studies of quantity, structure, space, and change.
• It is an essential tool applicable across natural science, engineering, medicine, finance, and social sciences.

Geometry

• Geometry studies the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
• This field includes shapes, sizes, relative positions of figures, and the properties of space.

Basic Geometric Concepts

• Point: A location in space, without dimension.
• Line: An infinitely long, straight path lacking width.
• Plane: A flat, two-dimensional surface extending infinitely.
• Angle: Measures the rotation between two lines or surfaces meeting at a common point.

Shapes

• Triangle: A polygon featuring three edges and three vertices.
• Square: A quadrilateral defined by four equal sides and four right angles.
• Circle: A set of points equidistant from a center point on a plane.

Algebra

• Algebra deals with symbols and the rules to manipulate them.
• It generalizes arithmetic, using letters and symbols to represent numbers, quantities, and relationships.

Basic Algebraic Concepts

• Variable: A symbol, often a letter, representing an unknown value.
• Expression: A combination of variables, numbers, and operations.
• Equation: A statement asserting the equality of two expressions.
• Coefficient: A number multiplied by a variable.
• Constant: A fixed, unchanging value.

Operations

• Addition: Combines numbers or terms.
• Subtraction: Finds the difference between numbers or terms.
• Multiplication: Repeated addition or scaling of numbers or terms.
• Division: Splits numbers or terms into equal parts.

Pythagoras Theorem

• The Pythagorean Theorem is a core relationship in Euclidean geometry concerning the sides of a right triangle.
• The theorem states the square of the hypotenuse's length equals the sum of the squares of the other two sides' lengths.

Formula

• Given a right triangle with legs of length a and b, and a hypotenuse of length c: a² + b² = c²

Applications

• Find the length of an unknown side in a right triangle given the other two sides.
• Confirm if a triangle is a right triangle.
• Facilitates construction of right angles and geometric computations.

Simultaneous Equations

• Simultaneous equations involve two or more equations containing multiple variables.
• Solutions satisfy all equations in the system at once.

Methods to Solve

• Substitution: Solve for one variable in one equation, then substitute that expression into another equation.
• Elimination: Add or subtract multiples of equations to eliminate a variable.
• Graphing: Plot each equation and find intersection points, which represent solutions.
• Matrix Methods: Employ matrix operations to solve linear equation systems, especially for larger systems.

Linear Equations

• Equations where the highest power of any variable is 1.
• Graphically represented by straight lines.

Non-linear Equations

• Equations with terms where variable powers are not 1, or containing complex functions.
• Can represent curves or non-linear shapes when graphed.

Solving by Substitution

• Isolate one variable in one equation in terms of the other.
• Replace this expression into the other equation.
• Solve the new equation for the remaining single variable.
• Substitute the found variable's value back into an initial equation to solve for the other variable.

Solving by Elimination

• Adjust one or both equations so that one variable has the same or opposite coefficients.
• Combine equations via addition or subtraction, eliminating one variable.
• Solve the remaining equation for the single variable.
• Input the solved variable's value into a starting equation to determine the value of the other variable.