TR-09: The Pythagorean Theorem (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial introduces the Pythagorean Theorem, a fundamental principle in trigonometry used to find relationships between the sides of right triangles. Understanding this theorem is essential for solving various geometric problems, including finding the length of the third side of a triangle and determining whether a triangle is a right triangle based on its side lengths.

Step 1: Understanding the Pythagorean Theorem

• The theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
• The formula is represented as:

  a² + b² = c²
  

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

Step 2: Finding the Length of the Third Side

To find the length of the unknown side in a right triangle, follow these steps:

1. Identify the lengths of the two known sides (a and b).
2. Use the Pythagorean Theorem formula:
   • If you need to find the hypotenuse (c):

     c = √(a² + b²)
     

   • If you need to find one leg (a or b):

     a = √(c² - b²)
     b = √(c² - a²)
     

3. Substitute the known values into the equation and calculate.

Practical Tip

• Ensure that your calculator is set to the correct mode (degree/radian) if you're using trigonometric functions after applying the theorem.

Step 3: Confirming if a Triangle is a Right Triangle

To determine if you have a right triangle with known side lengths:

1. Assign the longest side as c (hypotenuse) and the other two sides as a and b.
2. Calculate using the Pythagorean Theorem:

   a² + b²
   

3. Compare the result with c²:
   • If they are equal, the triangle is a right triangle.
   • If not, it is not a right triangle.

Common Pitfall to Avoid

• Ensure that you correctly identify the hypotenuse as the longest side before performing calculations.

Conclusion

The Pythagorean Theorem is a powerful tool for solving problems involving right triangles. By mastering this theorem, you can easily find missing side lengths and determine the nature of triangles based on their side lengths. For further practice, consider exploring Pythagorean triples, which are sets of three positive integers that satisfy the theorem, and enhance your understanding through additional exercises.