Parallelograms and Rhombuses

This section explores the properties and area calculations for parallelograms and rhombuses, which are important shapes in geometria płaska -- pole czworokąta sprawdzian liceum.

The key formulas introduced are:

Definition: Area of a parallelogram: A = base × height Definition: Area of a rhombus: A = (diagonal1 × diagonal2) / 2

Several example problems demonstrate how to apply these formulas in different scenarios:

1. Calculating parallelogram area given side lengths and included angle
2. Finding parallelogram height given area and base length
3. Determining rhombus area using diagonal lengths

Example: For a parallelogram with sides 6 cm and 13 cm and an included angle of 30°, the area can be calculated as: A = 6 × 13 × sin(30°) = 39 cm²

The section also covers more advanced concepts:

• Relationship between parallelogram heights
• Properties of rhombuses (diagonals bisect angles)
• Special cases where parallelograms become rhombuses

Highlight: If the diagonals of a parallelogram bisect its angles, then the parallelogram is a rhombus.

These problems help students develop a deeper understanding of wzór na pole dowolnego czworokąta and how to apply it to various parallelogram and rhombus configurations.

Page 4: General Quadrilateral Areas

Advanced applications of Geometria płaska -- pole czworokąta sprawdzian concepts focusing on diagonal-based area calculations.

Highlight: Area calculations using diagonal lengths and the angle between them.

Example: Quadrilateral with diagonals d₁ = 10, d₂ = 6, and angle α = 30°.

Rectangles and Squares

This section focuses on calculating the areas of rectangles and squares, which are fundamental quadrilaterals in geometria płaska czworokąty.

The key formulas introduced are:

Definition: Area of a rectangle: A = length × width Definition: Area of a square: A = side length²

Several example problems are provided to illustrate how to apply these formulas:

1. Finding the area of a square given the radius of inscribed and circumscribed circles
2. Calculating the ratio of areas between two squares with specific angle relationships
3. Determining rectangle dimensions given perimeter and area constraints

Example: For a rectangle with perimeter 18 cm and area 20 cm², the dimensions can be found by solving the system of equations: 2(l+w) = 18 and l×w = 20

The section also covers more advanced concepts related to rectangles:

• Relationship between rectangle diagonals and area
• Properties of rectangles inscribed in circles
• Dividing rectangles into smaller rectangles and analyzing area ratios

Highlight: The diagonals of a rectangle divide it into four triangles of equal area.

These problems help develop a deeper understanding of pole prostokąta i kwadratu beyond just applying basic formulas.