Applying the Pythagorean Theorem to Triangle QRS

Oct 13, 2025 by miodeya.com 137 views

Introduction

When it comes to geometry, one of the most fundamental concepts is the right-angled triangle. A right-angled triangle is a triangle with one angle that is 90 degrees. In this article, we will be discussing a specific right-angled triangle, known as triangle QRS. The legs of this triangle each have a length of 10 units, and we are tasked with finding the length of the hypotenuse.

What is a Hypotenuse?

Before we dive into the solution, let's first understand what a hypotenuse is. The hypotenuse is the longest side of a right-angled triangle, opposite the right angle. It is the side that is always opposite the right angle and is the longest side of the triangle.

The Pythagorean Theorem

To find the length of the hypotenuse, we can use the Pythagorean theorem. The Pythagorean theorem is a fundamental concept in geometry that states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Mathematically, this can be expressed as:

a^2 + b^2 = c^2

where a and b are the lengths of the two sides that form the right angle, and c is the length of the hypotenuse.

Applying the Pythagorean Theorem to Triangle QRS

Now that we have a good understanding of the Pythagorean theorem, let's apply it to triangle QRS. We know that the legs of the triangle each have a length of 10 units, so we can substitute these values into the equation:

a^2 + b^2 = c^2 10^2 + 10^2 = c^2 100 + 100 = c^2 200 = c^2

Solving for the Hypotenuse

Now that we have the equation 200 = c^2, we can solve for the length of the hypotenuse. To do this, we need to take the square root of both sides of the equation:

c = √200 c = √(100 × 2) c = √100 × √2 c = 10√2

Conclusion

In conclusion, the length of the hypotenuse of triangle QRS is 10√2 units. This is the correct answer, and it can be verified by using the Pythagorean theorem.

Why is the Pythagorean Theorem Important?

The Pythagorean theorem is a fundamental concept in geometry that has numerous applications in various fields, including physics, engineering, and architecture. It is used to calculate the length of the hypotenuse of a right-angled triangle, which is essential in many real-world applications.

Real-World Applications of the Pythagorean Theorem

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

Building Design: Architects use the Pythagorean theorem to calculate the length of the hypotenuse of a right-angled triangle, which is essential in designing buildings and structures.
Physics: Physicists use the Pythagorean theorem to calculate the length of the hypotenuse of a right-angled triangle, which is essential in understanding the motion of objects.
Engineering: Engineers use the Pythagorean theorem to calculate the length of the hypotenuse of a right-angled triangle, which is essential in designing bridges and other structures.

Common Mistakes to Avoid

When using the Pythagorean theorem, there are several common mistakes to avoid, including:

Not using the correct formula: Make sure to use the correct formula, which is a^2 + b^2 = c^2.
Not substituting the correct values: Make sure to substitute the correct values into the equation.
Not solving for the correct variable: Make sure to solve for the correct variable, which is the length of the hypotenuse.

Conclusion

In conclusion, the Pythagorean theorem is a fundamental concept in geometry that has numerous applications in various fields. It is used to calculate the length of the hypotenuse of a right-angled triangle, which is essential in many real-world applications. By understanding the Pythagorean theorem and avoiding common mistakes, we can use it to solve a wide range of problems.

Final Answer

The final answer is: $\boxed{10\sqrt{2}}$

Introduction

The Pythagorean theorem is a fundamental concept in geometry that has numerous applications in various fields. In our previous article, we discussed the basics of the Pythagorean theorem and how to use it to calculate the length of the hypotenuse of a right-angled triangle. In this article, we will be answering some of the most frequently asked questions about the Pythagorean theorem.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in geometry that states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Q: How do I use the Pythagorean theorem?

A: To use the Pythagorean theorem, you need to substitute the lengths of the two sides that form the right angle into the equation a^2 + b^2 = c^2, where a and b are the lengths of the two sides and c is the length of the hypotenuse.

Q: What is the formula for the Pythagorean theorem?

A: The formula for the Pythagorean theorem is a^2 + b^2 = c^2, where a and b are the lengths of the two sides that form the right angle and c is the length of the hypotenuse.

Q: Can I use the Pythagorean theorem to calculate the length of the hypotenuse of a triangle that is not a right-angled triangle?

A: No, the Pythagorean theorem can only be used to calculate the length of the hypotenuse of a right-angled triangle.

Q: What is the difference between the Pythagorean theorem and the Pythagorean identity?

A: The Pythagorean theorem is a formula that is used to calculate the length of the hypotenuse of a right-angled triangle, while the Pythagorean identity is a mathematical statement that is used to prove the Pythagorean theorem.

Q: Can I use the Pythagorean theorem to calculate the length of the sides of a triangle that is not a right-angled triangle?

A: No, the Pythagorean theorem can only be used to calculate the length of the hypotenuse of a right-angled triangle. To calculate the length of the sides of a triangle that is not a right-angled triangle, you need to use other mathematical formulas and techniques.

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

A: The Pythagorean theorem has numerous real-world applications, including building design, physics, engineering, and architecture.

Q: Can I use the Pythagorean theorem to calculate the length of the hypotenuse of a triangle that has a negative length?

A: No, the Pythagorean theorem can only be used to calculate the length of the hypotenuse of a triangle that has a positive length.

Q: What is the significance of the Pythagorean theorem in mathematics?

A: The Pythagorean theorem is a fundamental concept in mathematics that has numerous applications in various fields. It is used to calculate the length of the hypotenuse of a right-angled triangle, which is essential in many real-world applications.

Q: Can I use the Pythagorean theorem to calculate the length of the hypotenuse of a triangle that is a right-angled triangle but has a negative length?

A: No, the Pythagorean theorem can only be used to calculate the length of the hypotenuse of a right-angled triangle that has a positive length.

Conclusion

In conclusion, the Pythagorean theorem is a fundamental concept in geometry that has numerous applications in various fields. By understanding the Pythagorean theorem and its applications, we can use it to solve a wide range of problems.

Final Answer

The final answer is: $\boxed{10\sqrt{2}}$