Teorema Pythagoras [Part 3] - Menentukan Jenis Segitiga dan Tripel Pythagoras

Introduction

This tutorial will guide you through the concepts of the Pythagorean theorem, specifically focusing on determining types of triangles and understanding Pythagorean triples. This knowledge is crucial for grade 8 mathematics and can be particularly useful in practical applications like geometry and construction.

Step 1: Understanding the Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Important Formula

• For a right triangle with sides a, b, and hypotenuse c:

  c² = a² + b²
  

Practical Advice

• Always identify the right triangle first.
• Label the sides properly to avoid confusion.
• Use this theorem to determine if a triangle is right-angled by checking if the equation holds true.

Step 2: Identifying Types of Triangles

Triangles can be classified based on their angles and side lengths. The types include:

• Acute Triangle: All angles are less than 90 degrees.
• Right Triangle: One angle is exactly 90 degrees.
• Obtuse Triangle: One angle is greater than 90 degrees.

Practical Advice

• Use the Pythagorean theorem to verify if a triangle is a right triangle.
• Measure the angles with a protractor if necessary to classify the triangle accurately.

Step 3: Understanding Pythagorean Triples

A Pythagorean triple consists of three positive integers a, b, and c, such that they satisfy the Pythagorean theorem. Common examples include:

• (3, 4, 5)
• (5, 12, 13)
• (8, 15, 17)

Finding Pythagorean Triples

• You can generate Pythagorean triples using the formula:

  a = m² - n²
  b = 2mn
  c = m² + n²
  

  where m and n are positive integers, and m > n.

Practical Advice

• Check if the numbers satisfy the Pythagorean theorem to confirm they are a triple.
• Use these triples in real-world applications, such as construction, where right angles are common.

Step 4: Practical Application of the Theorem

In real-world scenarios, the Pythagorean theorem is often used to determine the dimensions of objects, such as frames or structures. For example, consider a window frame where:

• Length = 144 units
• Height = 42 units
• Diagonal = 150 units

Verification Process

• Apply the Pythagorean theorem:

  150² = 144² + 42²
  22,500 = 20,736 + 1,764
  22,500 = 22,500
  

• This confirms that the window forms a right triangle, validating the use of the theorem.

Conclusion

Understanding the Pythagorean theorem, types of triangles, and Pythagorean triples is essential in mathematics, especially in geometry. By mastering these concepts, you can solve various practical problems and improve your mathematical reasoning skills.

Next, practice identifying and calculating different triangles and their properties to reinforce your understanding.