TR-09Z: Proof of Pythagorean Theorem (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial provides a comprehensive guide to understanding the Pythagorean Theorem through a visual and graphical proof presented in the video by Dennis F. Davis. The Pythagorean Theorem is a fundamental principle in trigonometry and geometry, essential for solving problems involving right triangles. This guide will break down the proof into clear, actionable steps to enhance your understanding.

Step 1: Understanding the Basics of 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.
• The formula can be expressed as:

  a² + b² = c²
  

  where:
  • a and b are the lengths of the legs of the triangle,
  • c is the length of the hypotenuse.

Step 2: Visualizing the Theorem

• Create a right triangle on a piece of paper or use a drawing tool.
• Label the sides:
  • Side a (one leg)
  • Side b (the other leg)
  • Side c (the hypotenuse)
• Draw squares on each side of the triangle:
  • A square on side a will have an area of a².
  • A square on side b will have an area of b².
  • A square on side c will have an area of c².

Step 3: Constructing the Proof

• Use the squares drawn in Step 2 to visually demonstrate the theorem:
  • The combined area of the squares on sides a and b should equal the area of the square on side c.
• To illustrate this:
  • Cut out the squares from the drawing.
  • Rearrange the pieces to show how the two smaller squares fit into the larger square.

Step 4: Exploring Real-World Applications

• Discuss practical examples where the Pythagorean Theorem applies:
  • Architecture: Ensuring that structures are built at right angles.
  • Navigation: Calculating the shortest distance between two points.
  • Construction: Determining the correct lengths of materials needed for projects.

Conclusion

Understanding the Pythagorean Theorem through visual proof enhances comprehension and application in various fields. By following these steps, you can grasp the theorem's significance and how to apply it effectively. For further exploration, consider practicing with different right triangles and using the theorem in real-world scenarios.