Introduction

Relevance of the Theme

The Pythagorean Theorem, although seemingly simple, is a fundamental basis for Geometry, a discipline that studies the properties of space, shapes, and sizes. The theorem allows the understanding of the relationship between the sides of a right triangle, a vital concept in many aspects of Mathematics, Physics, Technical Drawing, and even in everyday applications such as measuring land and building furniture.

Contextualization

In the 9th-grade Mathematics curriculum of Elementary School, the Pythagorean Theorem is a crucial step before delving into the study of similarities and trigonometry. This is where the learning of geometric concepts deepens. The theorem represents a natural progression from studies on perimeters and areas, applying them to triangles with right angles. Without it, the understanding of these topics and of the right triangle itself would be incomplete.

Furthermore, the theorem gives us the first experience of how Mathematics can be used to represent the physical world - a central characteristic of many studies in Applied Mathematics. From this theorem, we begin to predict and describe the relationships and characteristics of the objects and phenomena around us.

Theoretical Development

Components

• Right Triangle: It is a special type of triangle with a 90-degree angle, known as a right angle. The other two angles are acute, meaning they are less than 90 degrees. The right triangle is the focal point of the Pythagorean Theorem, where the relationships between the sides are studied.
• Hypotenuse: It is the longest side in a right triangle and is directly opposite the right angle. In the calculation using the Pythagorean Theorem, the hypotenuse is often identified as 'c'.
• Catheti: They are the two shorter sides of a right triangle that form the right angle. In the calculation using the Pythagorean Theorem, the catheti are often identified as 'a' and 'b'.

Key Terms

• Pythagorean Theorem: A mathematical statement that establishes that, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the catheti. Mathematically, it is expressed as c^2 = a^2 + b^2.
• Square of a Number: It is the result of multiplying a number by itself. Symbol: x^2.
• Sum: Mathematical operation of addition, which results in the total of two quantities.

Examples and Cases

• Example 1: If we have a right triangle with one cathetus measuring 3 units and the other cathetus measuring 4 units, we can use the Pythagorean Theorem to find the length of the hypotenuse. According to the theorem, the square of the hypotenuse is equal to the sum of the squares of the catheti. Therefore, the square of the hypotenuse is 3^2 + 4^2, which is 25. The square root of 25 is 5, so the hypotenuse measures 5 units.

• Example 2: If we have a right triangle with a hypotenuse measuring 10 units and one cathetus measuring 6 units, we can use the Pythagorean Theorem to find the length of the other cathetus. According to the theorem, the length of the cathetus squared is equal to the square of the hypotenuse minus the square of the other cathetus. Therefore, the square of the other cathetus is 10^2 - 6^2, which is 100-36=64. The square root of 64 is 8, so the second cathetus measures 8 units.

• Case: Consider a square in the shape of a right triangle. You know that the distance from the starting point where you are to the first landmark measures 30 meters and you want to know the distance to the next landmark. You realize that the starting points, you and the first landmark, and the next landmark form a right triangle. Therefore, you can use the Pythagorean Theorem to solve the problem and find out that the distance to the next landmark is 40 meters.

Detailed Summary

Key Points

• Understanding the structure and nomenclature of a right triangle: The Pythagorean Theorem can only be applied to right triangles - triangles that contain a right angle (90 degrees) and whose sides are catheti and the hypotenuse.
• Understanding the relationship between the sides of a right triangle: The sides of a right triangle are related through the Pythagorean Theorem - the sum of the squares of the catheti is equal to the square of the hypotenuse. This relationship is symbolically represented as c^2 = a^2 + b^2.
• Familiarity with the concept of the square of a number: The calculation of the Pythagorean Theorem involves the operation of squaring a number. That is, a number multiplied by itself (x^2).

Conclusions

• Ability to apply the Pythagorean Theorem to solve problems: Through the understanding of the concepts and formulas of the Pythagorean Theorem, it is possible to solve a series of practical problems, such as calculating the length of an unknown side of a right triangle.
• Importance of practical applications of mathematical theory: Through the Pythagorean Theorem, students can appreciate the applicability of mathematics in solving real-world problems, such as measuring land or designing furniture.

Exercises

1. Application Exercise of the Pythagorean Theorem: A building has a staircase that forms a 30-degree angle with the ground. The staircase reaches a window that is 10 meters high. How extensive is the staircase?

2. Right Triangle Identification Exercise: Given three line segments measuring 4 cm, 8 cm, and 10 cm, determine if it is possible to form a right triangle with these line segments and explain why.

3. Problem-solving Exercise with the Pythagorean Theorem:

   Carlos wants to build a rectangular fence for his dog in the backyard. He has one side of the square measuring 6 meters and wants the diagonal of the square to measure exactly 10 meters (from corner to corner). Is this measurement sufficient for his plan? Use the Pythagorean Theorem to help him find out.