TR 10: Pythagorean Triples (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial covers the concept of Pythagorean Triples, which are sets of three integers that satisfy the Pythagorean theorem. Understanding these triples is essential in various fields such as mathematics, engineering, and architecture. We will also explore two practical applications of the Pythagorean theorem: checking if a corner is square and constructing a square corner.

Step 1: Understanding Pythagorean Triples

Pythagorean Triples consist of three positive integers (a), (b), and (c) that satisfy the equation:

[ a^2 + b^2 = c^2 ]

Example of a Pythagorean Triple

• The simplest example is the triple (3, 4, 5):
  • Calculate (3^2 + 4^2):
    • (3^2 = 9)
    • (4^2 = 16)
    • (9 + 16 = 25)
  • (5^2 = 25), confirming that (3, 4, 5) is a Pythagorean Triple.

Common Pythagorean Triples

• (5, 12, 13)
• (8, 15, 17)

Practical Tip

To generate more Pythagorean Triples, you can use the formula:

• For any integers (m) and (n) (where (m > n > 0)):
  • (a = m^2 - n^2)
  • (b = 2mn)
  • (c = m^2 + n^2)

Step 2: Checking if a Corner is Square

You can use the Pythagorean theorem to determine if a corner forms a right angle.

Steps to Check

1. Measure the two shorter sides of the corner (let's call them (a) and (b)).
2. Calculate (c) using the formula: [ c = \sqrt{a^2 + b^2} ]
3. Measure the diagonal (hypotenuse) of the corner and compare it to your calculated (c):
   • If the diagonal matches (c), the corner is square.
   • If not, adjustments may be necessary.

Common Pitfalls

• Ensure measurements are taken accurately, and the tape measure is straight.

Step 3: Constructing a Square Corner

You can construct a perfect right angle using the 3-4-5 method.

Steps to Construct

1. Mark a Point: Start at a point where you want to create the corner.
2. Measure 3 Units: From the point, measure and mark a point 3 units away.
3. Measure 4 Units: From the same starting point, measure and mark a point 4 units away.
4. Check the Diagonal: Measure the diagonal between the two marked points:
   • If the distance is 5 units, you have a square corner.

Real-World Application

This method is widely used in construction to ensure walls and foundations are laid out correctly.

Conclusion

Pythagorean Triples are fundamental in both theory and practical applications, such as ensuring corners are square and constructing right angles. By mastering these concepts and techniques, you can enhance your mathematical skills and apply them in real-world situations.

Next, consider exploring additional problems and drills related to Pythagorean triples to further solidify your understanding. For more advanced topics, check out the next video in the series.