Characteristic Triangles: 30-60-90 and 45-45-90

This page provides essential information about two important trójkąty charakterystyczne (characteristic triangles): the 30-60-90 triangle and the 45-45-90 triangle. These triangles have specific angle measurements and unique side length relationships that make them valuable in geometry.

The document begins by reminding us of a fundamental property of triangles: the sum of all angles in a triangle is always 180°. It also provides the general formula for calculating the area of a triangle: A = (base × height) / 2.

For the 45-45-90 triangle, also known as an isosceles right triangle, the page shows a diagram illustrating its shape and side length relationships. In this triangle, two sides are equal in length, and the hypotenuse is √2 times the length of a leg.

Definition: A 45-45-90 triangle is a right triangle with two 45° angles and one 90° angle. It is also an isosceles triangle because two of its sides are equal in length.

The 30-60-90 triangle, also called a special right triangle, is depicted with its unique side length ratios. The shortest side (opposite to the 30° angle) has a length of a, the hypotenuse (opposite to the 90° angle) has a length of 2a, and the remaining side (opposite to the 60° angle) has a length of a√3.

Highlight: The 30-60-90 triangle is actually half of an equilateral triangle. This relationship explains its unique side length ratios.

The document provides an alternative notation for the 30-60-90 triangle, where the shortest side is labeled as a/2 instead of a. This notation can be useful in certain problem-solving scenarios.

Example: In a 30-60-90 triangle with the shortest side length of 2 units, the hypotenuse would be 4 units, and the remaining side would be 2√3 units.

Finally, the page reiterates the general formula for calculating the area of a triangle: Area = (base × height) / 2. This formula applies to all triangles, including the characteristic triangles discussed on this page.

Vocabulary:

• Trójkąt 30 60 90: A right triangle with angles of 30°, 60°, and 90°.
• Trójkąt równoramienny: Isosceles triangle, where two sides are equal in length.
• Pole trójkąta: Area of a triangle.