Which Of The Following Is A Pythagorean Triple?A. (5, 12, 13) B. (5, 8, 10) C. (1, 1, 2) D. (3, 4, 6)

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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. In this article, we will explore which of the given options is a Pythagorean triple.

What are Pythagorean Triples?

A Pythagorean triple is a set of three positive integers a, b, and c, such that a^2 + b^2 = c^2. This equation is known as the Pythagorean theorem, and it describes the relationship between the lengths of the sides of a right-angled triangle. The Pythagorean theorem is a fundamental concept in geometry and trigonometry, and it has numerous applications in various fields, including physics, engineering, and computer science.

Properties of Pythagorean Triples

Pythagorean triples have several interesting properties. For example, the ratio of the lengths of the sides of a right-angled triangle is always a rational number. This means that the ratio of the lengths of the sides can be expressed as a fraction of two integers. Additionally, the sum of the squares of the lengths of the legs of a right-angled triangle is equal to the square of the length of the hypotenuse.

Examples of Pythagorean Triples

There are many examples of Pythagorean triples. Some of the most well-known examples include:

  • (3, 4, 5)
  • (5, 12, 13)
  • (8, 15, 17)
  • (7, 24, 25)

These triples are often used in geometry and trigonometry to illustrate the Pythagorean theorem and to solve problems involving right-angled triangles.

Which of the Given Options is a Pythagorean Triple?

Now that we have discussed the properties and examples of Pythagorean triples, let's examine the given options to determine which one is a Pythagorean triple.

  • A. (5, 12, 13)
  • B. (5, 8, 10)
  • C. (1, 1, 2)
  • D. (3, 4, 6)

To determine which option is a Pythagorean triple, we need to check if the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

Checking Option A

Let's check if option A is a Pythagorean triple.

a^2 + b^2 = c^2 5^2 + 12^2 = 13^2 25 + 144 = 169 169 = 169

Since the equation is true, option A is a Pythagorean triple.

Checking Option B

Let's check if option B is a Pythagorean triple.

a^2 + b^2 = c^2 5^2 + 8^2 = 10^2 25 + 64 = 100 100 = 100

Since the equation is true, option B is a Pythagorean triple.

Checking Option C

Let's check if option C is a Pythagorean triple.

a^2 + b^2 = c^2 1^2 + 1^2 = 2^2 1 + 1 = 4 4 = 4

Since the equation is true, option C is a Pythagorean triple.

Checking Option D

Let's check if option D is a Pythagorean triple.

a^2 + b^2 = c^2 3^2 + 4^2 = 6^2 9 + 16 = 36 25 = 36

Since the equation is not true, option D is not a Pythagorean triple.

Conclusion

In conclusion, the Pythagorean triples are sets of three positive integers a, b, and c, such that a^2 + b^2 = c^2. We have discussed the properties and examples of Pythagorean triples and have examined the given options to determine which one is a Pythagorean triple. The options that are Pythagorean triples are:

  • A. (5, 12, 13)
  • B. (5, 8, 10)
  • C. (1, 1, 2)

These triples are often used in geometry and trigonometry to illustrate the Pythagorean theorem and to solve problems involving right-angled triangles.

References

  • "Pythagorean Theorem" by Math Open Reference
  • "Pythagorean Triples" by Wolfram MathWorld
  • "Geometry and Trigonometry" by Khan Academy

Frequently Asked Questions

  • 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.
  • Q: What are some examples of Pythagorean triples? A: Some examples of Pythagorean triples include (3, 4, 5), (5, 12, 13), and (8, 15, 17).
  • Q: How do I determine if a set of numbers is a Pythagorean triple? A: To determine if a set of numbers is a Pythagorean triple, you need to check if the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
    Pythagorean Triples Q&A ==========================

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. This equation is known as the Pythagorean theorem, and it describes the relationship between the lengths of the sides of a right-angled triangle.

Q: What are some examples of Pythagorean triples?

A: Some examples of Pythagorean triples include:

  • (3, 4, 5)
  • (5, 12, 13)
  • (8, 15, 17)
  • (7, 24, 25)

These triples are often used in geometry and trigonometry to illustrate the Pythagorean theorem and to solve problems involving right-angled triangles.

Q: How do I determine if a set of numbers is a Pythagorean triple?

A: To determine if a set of numbers is a Pythagorean triple, you need to check if the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. This can be done using the following equation:

a^2 + b^2 = c^2

If the equation is true, then the set of numbers is a Pythagorean triple.

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

A: Pythagorean triples have numerous real-world applications, including:

  • Building design: Pythagorean triples are used to calculate the lengths of the sides of right-angled triangles in building design.
  • Physics: Pythagorean triples are used to calculate the distances and velocities of objects in motion.
  • Engineering: Pythagorean triples are used to calculate the lengths of the sides of right-angled triangles in engineering design.
  • Computer graphics: Pythagorean triples are used to create 3D models and animations.

Q: Can I generate my own Pythagorean triples?

A: Yes, you can generate your own Pythagorean triples using the following formula:

a = m^2 - n^2 b = 2mn c = m^2 + n^2

where m and n are positive integers.

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 if the equation a^2 + b^2 = c^2 is true before assuming that a set of numbers is a Pythagorean triple.
  • Not using the correct formula to generate Pythagorean triples.
  • Not checking if the set of numbers is a Pythagorean triple before using it in a calculation.

Q: Can I use Pythagorean triples to solve problems involving oblique triangles?

A: Yes, you can use Pythagorean triples to solve problems involving oblique triangles. However, you will need to use the law of cosines or the law of sines to calculate the lengths of the sides of the triangle.

Q: What are some advanced topics related to Pythagorean triples?

A: Some advanced topics related to Pythagorean triples include:

  • Pythagorean quadruples: These are sets of four positive integers a, b, c, and d, such that a^2 + b^2 = c^2 and c^2 + d^2 = e^2.
  • Pythagorean pentuples: These are sets of five positive integers a, b, c, d, and e, such that a^2 + b^2 = c^2 and c^2 + d^2 = e^2.
  • Pythagorean triples with irrational numbers: These are sets of three positive integers a, b, and c, such that a^2 + b^2 = c^2, where c is an irrational number.

Q: Where can I learn more about Pythagorean triples?

A: You can learn more about Pythagorean triples by:

  • Reading books on geometry and trigonometry.
  • Taking online courses on geometry and trigonometry.
  • Practicing problems involving Pythagorean triples.
  • Joining online communities and forums related to geometry and trigonometry.

References

  • "Pythagorean Theorem" by Math Open Reference
  • "Pythagorean Triples" by Wolfram MathWorld
  • "Geometry and Trigonometry" by Khan Academy

Frequently Asked Questions

  • 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.
  • Q: What are some examples of Pythagorean triples? A: Some examples of Pythagorean triples include (3, 4, 5), (5, 12, 13), and (8, 15, 17).
  • Q: How do I determine if a set of numbers is a Pythagorean triple? A: To determine if a set of numbers is a Pythagorean triple, you need to check if the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.