Symmetry Line and Circle Equations

This page focuses on deriving the equation of a symmetry line for a line segment and introduces circle equations in a coordinate system.

To find the equation of a symmetry line for a line segment, follow these steps:

1. Determine the midpoint of the segment
2. Calculate the slope of the original segment
3. Find the slope of the symmetry line
4. Use the midpoint and slope to derive the symmetry line equation

Vocabulary: The symetralna odcinka (symmetry line of a segment) is a perpendicular line that passes through the midpoint of the segment.

The page presents two important principles:

1. Parallelism Principle: k || L ⇔ A₁B₂ - A₂B₁ = 0
2. Perpendicularity Principle: k ⊥ L ⇔ A₁A₂ + B₁B₂ = 0

These principles are crucial for analyzing relationships between lines in a coordinate system.

Definition: The równanie okręgu (circle equation) is presented in two forms:

1. Canonical form: (x - a)² + (y - b)² = r²
2. General form: x² + y² - 2ax - 2by + c = 0

Highlight: The radius of a circle can be calculated using the formula r² = a² + b² - c, where r > 0.

The page concludes with a system of equations representing two lines: L₁: A₁x + B₁y + C₁ = 0 L₂: A₂x + B₂y + C₂ = 0

This system is useful for finding intersection points of lines and solving various geometric problems in the coordinate system.

Vector Coordinates and Line Equations

This page introduces fundamental concepts related to współrzędne wektora (vector coordinates) and line equations in a coordinate system.

The formula for vector coordinates is presented as AB = [x₂ - x₁, y₂ - y₁]. This allows for calculating the horizontal and vertical components of a vector between two points.

Definition: The długość wektora wzór (vector length formula) is given by |AB| = √((x₂ - x₁)² + (y₂ - y₁)²), which calculates the straight-line distance between two points.

For finding the midpoint of a line segment, the formula (x₁ + x₂)/2, (y₁ + y₂)/2 is provided. This is useful for various geometric calculations.

The page also covers the slope formula, expressed as a = (y₂ - y₁)/(x₂ - x₁) = tan α. This is crucial for understanding the inclination of lines in the coordinate system.

Example: For parallel lines, y = a₁x + b₁ || y = a₂x + b₂, the condition a₁ = a₂ must be satisfied.

Highlight: Perpendicular lines have a special relationship between their slopes: a₁a₂ = -1 (or a₂ = -1/a₁).

The page concludes with examples of perpendicular lines, such as y = √3x + 3 ⊥ y = -2x - 6 and y = -7x + 1 ⊥ y = 1/7x - √3, demonstrating the application of the perpendicularity condition.