Communicate And Justify: Glen Found The Length Of The Hypotenuse Of A Right Triangle Using A 2 + B 2 \sqrt{a^2+b^2} A 2 + B 2 . Gigi Used ( A + B ) 2 \sqrt{(a+b)^2} ( A + B ) 2 . Who Is Correct? Explain.

Mar 12, 2025 by ADMIN 208 views

**Communicate and Justify: A Mathematical Debate**

Introduction

In the world of mathematics, communication and justification are crucial skills that enable individuals to convey their ideas effectively and defend their arguments. A recent debate between Glen and Gigi on the length of the hypotenuse of a right triangle highlights the importance of these skills. In this article, we will examine the methods used by Glen and Gigi and determine who is correct.

The Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right triangle. The theorem states that 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). Mathematically, this can be expressed as:

c^2 = a^2 + b^2

This theorem is a powerful tool for solving problems involving right triangles and has numerous applications in various fields, including physics, engineering, and architecture.

Glen's Method

Glen used the formula $\sqrt{a^2+b^2}$ to find the length of the hypotenuse. This formula is a direct application of the Pythagorean Theorem, where the square root of the sum of the squares of the lengths of the other two sides is taken to obtain the length of the hypotenuse. Glen's method is correct because it accurately represents the relationship between the sides of a right triangle.

Gigi's Method

Gigi, on the other hand, used the formula $\sqrt{(a+b)^2}$ . This formula appears to be a simple extension of the Pythagorean Theorem, where the sum of the lengths of the other two sides is squared and then taken as the square root. However, this method is incorrect because it does not accurately represent the relationship between the sides of a right triangle.

Justification

To justify why Gigi's method is incorrect, let's examine the formula $\sqrt{(a+b)^2}$ . When we expand the square of the sum of the lengths of the other two sides, we get:

(a+b)^2 = a^2 + 2ab + b^2

Taking the square root of this expression, we get:

√(a^2 + 2ab + b^2)

This expression is not equal to the length of the hypotenuse, which is given by the Pythagorean Theorem as:

c^2 = a^2 + b^2

Therefore, Gigi's method is incorrect because it does not accurately represent the relationship between the sides of a right triangle.

Conclusion

In conclusion, Glen's method of using the formula $\sqrt{a^2+b^2}$ to find the length of the hypotenuse is correct, while Gigi's method of using the formula $\sqrt{(a+b)^2}$ is incorrect. This debate highlights the importance of communication and justification in mathematics, where clear and accurate representation of ideas is crucial for effective problem-solving.

Real-World Applications

The Pythagorean Theorem has numerous real-world applications, including:

Physics: The theorem is used to calculate the distance traveled by an object under the influence of gravity.
Engineering: The theorem is used to design and build structures, such as bridges and buildings.
Architecture: The theorem is used to calculate the height of buildings and the distance between buildings.

Tips for Effective Communication and Justification

To communicate and justify mathematical ideas effectively, follow these tips:

Be clear and concise: Avoid using complex language or jargon that may confuse others.
Use visual aids: Diagrams, graphs, and charts can help to illustrate mathematical concepts and make them more accessible.
Provide evidence: Use mathematical proofs and examples to support your arguments.
Be open to feedback: Encourage others to question and challenge your ideas, and be willing to revise and improve your arguments.

Conclusion

Introduction

In our previous article, we examined the methods used by Glen and Gigi to find the length of the hypotenuse of a right triangle. We determined that Glen's method of using the formula $\sqrt{a^2+b^2}$ is correct, while Gigi's method of using the formula $\sqrt{(a+b)^2}$ is incorrect. In this article, we will answer some frequently asked questions (FAQs) related to this debate.

Q: What is the Pythagorean Theorem?

A: The Pythagorean Theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right triangle. The theorem states that 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). Mathematically, this can be expressed as:

c^2 = a^2 + b^2

Q: Why is Glen's method correct?

A: Glen's method is correct because it accurately represents the relationship between the sides of a right triangle. The formula $\sqrt{a^2+b^2}$ is a direct application of the Pythagorean Theorem, where the square root of the sum of the squares of the lengths of the other two sides is taken to obtain the length of the hypotenuse.

Q: Why is Gigi's method incorrect?

A: Gigi's method is incorrect because it does not accurately represent the relationship between the sides of a right triangle. The formula $\sqrt{(a+b)^2}$ is not equal to the length of the hypotenuse, which is given by the Pythagorean Theorem as:

c^2 = a^2 + b^2

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

A: The Pythagorean Theorem has numerous real-world applications, including:

Physics: The theorem is used to calculate the distance traveled by an object under the influence of gravity.
Engineering: The theorem is used to design and build structures, such as bridges and buildings.
Architecture: The theorem is used to calculate the height of buildings and the distance between buildings.

Q: How can I communicate and justify mathematical ideas effectively?

A: To communicate and justify mathematical ideas effectively, follow these tips:

Be clear and concise: Avoid using complex language or jargon that may confuse others.
Use visual aids: Diagrams, graphs, and charts can help to illustrate mathematical concepts and make them more accessible.
Provide evidence: Use mathematical proofs and examples to support your arguments.
Be open to feedback: Encourage others to question and challenge your ideas, and be willing to revise and improve your arguments.

Q: What are some common mistakes to avoid when using the Pythagorean Theorem?

A: Some common mistakes to avoid when using the Pythagorean Theorem include:

Not squaring the lengths of the sides: Make sure to square the lengths of the sides before adding them together.
Not taking the square root: Make sure to take the square root of the result to obtain the length of the hypotenuse.
Not checking for errors: Double-check your calculations to ensure that they are accurate.

Conclusion

In conclusion, the debate between Glen and Gigi highlights the importance of effective communication and justification in mathematics. By using clear and accurate language, providing evidence, and being open to feedback, we can convey our ideas effectively and defend our arguments. We hope that this Q&A article has provided valuable insights and tips for anyone interested in mathematics.