Special Right Triangles: 30-60-90 and 45-45-90

This page focuses on two special types of right triangles: the 30-60-90 triangle and the 45-45-90 triangle. These triangles have unique properties and ratios that make them particularly useful in geometry and trigonometry.

The 30-60-90 triangle is described as half of an equilateral triangle. Its side ratios are 1 : √3 : 2, corresponding to the shortest side (opposite to 30°), the side opposite to 60°, and the hypotenuse, respectively.

Definition: A trójkąt 30 60 90 is a right triangle with angles of 30°, 60°, and 90°, formed by bisecting an equilateral triangle.

The 45-45-90 triangle is presented as half of a square. Its two legs are equal, and the hypotenuse is √2 times the length of a leg.

Highlight: The side ratios in a 45-45-90 triangle are 1 : 1 : √2, representing the two equal legs and the hypotenuse.

An example problem is provided, involving a square-shaped plaza with an underground cable running along its diagonal. The problem demonstrates how to use the properties of a 45-45-90 triangle to calculate the area of the square given the length of its diagonal.

Example: Given a diagonal (cable length) of 10m, the side length of the square is calculated as 5√2 m, resulting in an area of 100 m².

The page also touches on the concept of zależności w trójkącie 30 60 90 (relationships in a 30-60-90 triangle) and zależności w trójkącie 45 45 90 (relationships in a 45-45-90 triangle), which are crucial for solving various geometric problems.

Equilateral Triangle Properties and Calculations

This page delves into the characteristics and formulas related to equilateral triangles. An equilateral triangle has three equal sides and three 60° angles, making it a unique and symmetrical shape in geometry.

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 length of a side. This formula is derived from the general triangle area formula using the height of the equilateral triangle.

The wzór na wysokość trójkąta równobocznego (formula for the height of an equilateral triangle) is given as h = (a√3) / 2. This formula is crucial for various calculations involving equilateral triangles.

Example: A problem is presented where the height of an equilateral triangle is 5 cm, and students are asked to calculate the side length. Using the height formula, the side length is determined to be 2√3 cm.

Highlight: The page emphasizes that in an equilateral triangle, the altitudes, angle bisectors, perpendicular bisectors, and medians all coincide and intersect at a single point, which divides each of these lines in a 2:1 ratio.

Another example problem demonstrates how to calculate the side length and height of an equilateral triangle given its area of 4√3 cm². This problem illustrates the practical application of the area and height formulas for equilateral triangles.

Vocabulary: Pole trójkąta równobocznego refers to the area of an equilateral triangle, while trójkąt równoboczny wzory encompasses all formulas related to equilateral triangles.