Right-Angled Triangles

This page focuses on the properties and formulas related to right-angled triangles.

The wzór na pole trójkąta prostokątnego (formula for the area of a right-angled triangle) is presented in multiple forms:

1. P = $1/2$ab, where 'a' and 'b' are the lengths of the two perpendicular sides.
2. P = $1/2$ch, where 'c' is the hypotenuse and 'h' is the height to the hypotenuse.

The Pythagorean theorem is highlighted: a² + b² = c², where 'c' is the hypotenuse and 'a' and 'b' are the other two sides.

Definition: Altitude (wysokość) - The perpendicular line segment from a vertex to the opposite side (or its extension) in a triangle.

Example: The formula for the altitude of a right-angled triangle is given as h = (xy) / c, where 'x' and 'y' are segments of the hypotenuse created by the altitude.

Highlight: The diameter of the circumcircle of a right-angled triangle is equal to the length of its hypotenuse.

Scalene Triangles and General Triangle Properties

This page covers scalene triangles and general properties applicable to all triangles.

For scalene triangles, the wzór na pole trójkąta (formula for the area of a triangle) is presented in various forms:

1. P = $1/2$ah, where 'a' is any side and 'h' is the corresponding height.
2. P = $1/2$bc sin α, where 'b' and 'c' are two sides and α is the angle between them.

The page also discusses the classification of triangles based on their angles:

• Acute triangle: All angles < 90°
• Right-angled triangle: One angle = 90°
• Obtuse triangle: One angle > 90°

Vocabulary: Scalene triangle (trójkąt różnoboczny) - a triangle with no two sides of equal length.

Highlight: In an acute triangle, the center of the circumcircle lies inside the triangle, while in an obtuse triangle, it lies outside.

General Triangle Formulas and Heron's Formula

This final page presents general formulas applicable to all triangles and introduces Heron's formula.

The wzór na pole trójkąta (formula for the area of a triangle) is given as P = pr, where 'p' is the semi-perimeter and 'r' is the inradius.

The relationship between the area (P), semi-perimeter $p$, and circumradius (R) is shown as: P = abc / (4R)

Highlight: Heron's formula for calculating the area of a triangle: P = √$p(p-a)(p-b)(p-c)$, where 'p' is the semi-perimeter and 'a', 'b', 'c' are the side lengths.

Definition: Semi-perimeter (półobwód) - Half the perimeter of a triangle, calculated as p = $a + b + c$ / 2.

Example: The pole trójkąta kalkulator (triangle area calculator) often uses Heron's formula as it only requires the lengths of the three sides to compute the area.

Equilateral and Isosceles Triangles

This page introduces the formulas and properties of equilateral and isosceles triangles.

For equilateral triangles, the wzór na pole trójkąta równobocznego (formula for the area of an equilateral triangle) is presented as P = (a² √3) / 4, where 'a' is the side length. The wzór na wysokość trójkąta równobocznego (formula for the height of an equilateral triangle) is given as h = (a √3) / 2.

For isosceles triangles, the wzór na pole trójkąta równoramiennego (formula for the area of an isosceles triangle) is shown as P = $1/2$ah, where 'a' is the base and 'h' is the height. An alternative formula using the angle between equal sides is also provided: P = $1/2$b² sin α.

Vocabulary: Equilateral triangle (trójkąt równoboczny) - a triangle with all sides equal in length.

Vocabulary: Isosceles triangle (trójkąt równoramienny) - a triangle with two sides of equal length.

Highlight: The relationship between the inradius $r$ and circumradius (R) of an equilateral triangle is shown as R = 2r.