Leon Verified That The Side Lengths 21, 28, And 35 Form A Pythagorean Triple Using This Procedure:Step 1: Find The Greatest Common Factor Of The Given Lengths: 7 Step 2: Divide The Given Lengths By The Greatest Common Factor: 3, 4, 5 Step 3: Verify

by ADMIN 251 views

What are Pythagorean Triples?

Pythagorean triples are sets of three positive integers a, b, and c, such that a^2 + b^2 = c^2. These triples are named after the ancient Greek philosopher and mathematician Pythagoras, who is credited with their discovery. Pythagorean triples have numerous applications in mathematics, physics, and engineering, and are a fundamental concept in geometry and trigonometry.

Leon's Procedure for Verifying Pythagorean Triples

Leon, a mathematician, has developed a procedure for verifying whether a given set of side lengths forms a Pythagorean triple. The procedure involves three steps:

Step 1: Find the Greatest Common Factor (GCF) of the Given Lengths

The first step in Leon's procedure is to find the greatest common factor (GCF) of the given side lengths. The GCF is the largest positive integer that divides each of the given numbers without leaving a remainder. In the case of the side lengths 21, 28, and 35, the GCF is 7.

Why is the GCF important?

The GCF is important because it allows us to simplify the given side lengths and make it easier to verify whether they form a Pythagorean triple. By dividing the given side lengths by the GCF, we can obtain a new set of side lengths that are more manageable and easier to work with.

Step 2: Divide the Given Lengths by the Greatest Common Factor

Once we have found the GCF, we can divide the given side lengths by the GCF to obtain a new set of side lengths. In the case of the side lengths 21, 28, and 35, dividing by 7 gives us the new side lengths 3, 4, and 5.

What do these new side lengths represent?

The new side lengths 3, 4, and 5 represent a simplified version of the original side lengths. These new side lengths are more manageable and easier to work with, and they can be used to verify whether the original side lengths form a Pythagorean triple.

Step 3: Verify the Pythagorean Triple

The final step in Leon's procedure is to verify whether the new side lengths form a Pythagorean triple. This involves checking whether the sum of the squares of the two shorter side lengths is equal to the square of the longest side length. In the case of the side lengths 3, 4, and 5, we have:

3^2 + 4^2 = 9 + 16 = 25 5^2 = 25

Since the sum of the squares of the two shorter side lengths (3 and 4) is equal to the square of the longest side length (5), we can conclude that the side lengths 3, 4, and 5 form a Pythagorean triple.

Conclusion

Leon's procedure for verifying Pythagorean triples is a simple and effective method for determining whether a given set of side lengths forms a Pythagorean triple. By finding the greatest common factor of the given side lengths, dividing the given side lengths by the GCF, and verifying the Pythagorean triple, we can determine whether a given set of side lengths is a Pythagorean triple.

Real-World Applications of Pythagorean Triples

Pythagorean triples have numerous real-world applications in mathematics, physics, and engineering. Some examples include:

  • Geometry and Trigonometry: Pythagorean triples are used to calculate distances and angles in geometry and trigonometry.
  • Physics: Pythagorean triples are used to calculate distances and velocities in physics.
  • Engineering: Pythagorean triples are used to calculate distances and angles in engineering.
  • Computer Science: Pythagorean triples are used in computer science to calculate distances and angles in graphics and game development.

Famous Pythagorean Triples

Some famous Pythagorean triples include:

  • 3, 4, 5: This is one of the most well-known Pythagorean triples and is often used as an example in mathematics and physics.
  • 5, 12, 13: This Pythagorean triple is often used in geometry and trigonometry to calculate distances and angles.
  • 7, 24, 25: This Pythagorean triple is often used in physics to calculate distances and velocities.

Conclusion

Q: What is a Pythagorean triple?

A: A Pythagorean triple is a set of three positive integers a, b, and c, such that a^2 + b^2 = c^2. These triples are named after the ancient Greek philosopher and mathematician Pythagoras, who is credited with their discovery.

Q: How do I find a Pythagorean triple?

A: There are several ways to find a Pythagorean triple. One method is to use Leon's procedure, which involves finding the greatest common factor of the given side lengths, dividing the given side lengths by the GCF, and verifying the Pythagorean triple.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest positive integer that divides each of the given numbers without leaving a remainder. In the case of the side lengths 21, 28, and 35, the GCF is 7.

Q: Why is the GCF important?

A: The GCF is important because it allows us to simplify the given side lengths and make it easier to verify whether they form a Pythagorean triple. By dividing the given side lengths by the GCF, we can obtain a new set of side lengths that are more manageable and easier to work with.

Q: How do I verify a Pythagorean triple?

A: To verify a Pythagorean triple, you need to check whether the sum of the squares of the two shorter side lengths is equal to the square of the longest side length. In the case of the side lengths 3, 4, and 5, we have:

3^2 + 4^2 = 9 + 16 = 25 5^2 = 25

Since the sum of the squares of the two shorter side lengths (3 and 4) is equal to the square of the longest side length (5), we can conclude that the side lengths 3, 4, and 5 form a Pythagorean triple.

Q: What are some real-world applications of Pythagorean triples?

A: Pythagorean triples have numerous real-world applications in mathematics, physics, and engineering. Some examples include:

  • Geometry and Trigonometry: Pythagorean triples are used to calculate distances and angles in geometry and trigonometry.
  • Physics: Pythagorean triples are used to calculate distances and velocities in physics.
  • Engineering: Pythagorean triples are used to calculate distances and angles in engineering.
  • Computer Science: Pythagorean triples are used in computer science to calculate distances and angles in graphics and game development.

Q: What are some famous Pythagorean triples?

A: Some famous Pythagorean triples include:

  • 3, 4, 5: This is one of the most well-known Pythagorean triples and is often used as an example in mathematics and physics.
  • 5, 12, 13: This Pythagorean triple is often used in geometry and trigonometry to calculate distances and angles.
  • 7, 24, 25: This Pythagorean triple is often used in physics to calculate distances and velocities.

Q: Can I use a calculator to find a Pythagorean triple?

A: Yes, you can use a calculator to find a Pythagorean triple. However, it's often more fun and educational to use a manual method, such as Leon's procedure, to find a Pythagorean triple.

Q: How do I use a calculator to find a Pythagorean triple?

A: To use a calculator to find a Pythagorean triple, you can simply enter the side lengths into the calculator and press the "Pythagorean theorem" button. The calculator will then calculate the value of the Pythagorean triple.

Q: What are some common mistakes to avoid when working with Pythagorean triples?

A: Some common mistakes to avoid when working with Pythagorean triples include:

  • Not checking for the greatest common factor: Make sure to check for the greatest common factor of the given side lengths before proceeding.
  • Not verifying the Pythagorean triple: Make sure to verify the Pythagorean triple by checking whether the sum of the squares of the two shorter side lengths is equal to the square of the longest side length.
  • Not using the correct formula: Make sure to use the correct formula for the Pythagorean theorem, which is a^2 + b^2 = c^2.

Conclusion

Pythagorean triples are an important concept in mathematics, physics, and engineering. By understanding Pythagorean triples and their applications, we can gain a deeper appreciation for the beauty and importance of mathematics in our everyday lives.