Isosceles and Equilateral Triangle Problems
This page presents a series of geometric problems focusing on isosceles and equilateral triangles. The problems cover various aspects of planimetry, including calculations of heights, side lengths, and areas of these special triangles.
Definition: An isosceles triangle is a triangle with two sides of equal length.
Definition: An equilateral triangle is a triangle with all three sides of equal length.
The first problem deals with an isosceles triangle ABC where the equal sides AC and BC have a length of 5 units, and the height CD is 2 units. The task is to determine the length of the base AB.
Example: In an isosceles triangle with equal sides of 5 units and a height of 2 units, the base length can be calculated using the Pythagorean theorem.
The second problem involves an isosceles triangle with a base of 6 units and equal sides of 5 units. The question asks for the length of the height drawn to the base.
Highlight: The height in an isosceles triangle bisects the base and forms two right triangles, simplifying calculations.
The third problem presents an isosceles triangle ABC with equal sides AC and BC of length 7 units and a base AB of 12 units. The task is to find the length of the height drawn from vertex C to the base.
Vocabulary: The height (or altitude) of a triangle is a line segment from a vertex perpendicular to the opposite side (or its extension).
The final problem on this page involves an equilateral triangle with side length 24√3. The question asks for the radius of the inscribed circle in this triangle.
Highlight: In an equilateral triangle, the radius of the inscribed circle is related to the side length by a specific ratio.
These problems demonstrate the application of wzory na trójkąt równoboczny (formulas for equilateral triangles) and wzory na trójkąt równoramienny (formulas for isosceles triangles) in solving geometric questions, which are common in planimetria zadania z rozwiązaniami (planimetry problems with solutions).
