Analytic Geometry Formulas and Concepts

This page provides a comprehensive overview of key formulas and concepts in analytic geometry, essential for solving Geometria zadania liceum pdf problems. The content covers vectors, lines, circles, and their relationships in the Cartesian plane.

Vocabulary: Analytic geometry combines algebra and geometry to solve problems in a coordinate system.

Vectors and Lines

The document begins with vector operations and line equations, fundamental to understanding geometric relationships in the coordinate plane.

Example: The vector AB is represented as [XB-XA, YB-YA], which forms the basis for many calculations in analytic geometry.

The distance formula between two points is given as: |AB| = √((xB-xA)² + (yB-yA)²)

This formula is crucial for calculating lengths in the Płaszczyzna kartezjańska wzory system.

Definition: The equation of a line passing through point A with slope 'a' is: y - yA = a(x - xA)

Angles and Perpendicularity

The document provides formulas for calculating angles between lines and determining perpendicularity:

Highlight: The angle between two lines is given by: tan α = |(A₂B₁ - A₁B₂) / (A₁A₂ + B₁B₂)|

Perpendicularity of lines is determined when: A₁A₂ + B₁B₂ = 0

These concepts are crucial for solving Geometria zadania liceum pdf Nowa Era problems involving line relationships.

Circles and Their Equations

The general equation of a circle is presented as: x² + y² - 2ax - 2by + c = 0

This equation is fundamental for solving problems related to Równanie okręgu in the Cartesian plane.

Example: The relative position of a line and a circle can be determined by comparing the distance 'd' between the line and the circle's center to the radius 'r':

• d > r: External line
• d = r: Tangent line
• d < r: Intersecting line

Distance Formulas

The document includes important distance formulas, including the Wzór na długość odcinka and the formula for the Odległość punktu od prostej:

d(P,L) = |Ax₀ + By₀ + C| / √(A² + B²)

This formula is essential for solving Odległość punktu od prostej zadania in various geometric contexts.

Highlight: The area of a triangle can be calculated using the formula: P = ½|a₁b₂ - a₂b₁| = ½|ad - bc|

These formulas and concepts form the core of Planimetria wzory and are crucial for mastering Geometria na płaszczyźnie kartezjańskiej zadania maturalne pdf.