Which Of The Following Sides From A Right Triangle A) 2,3,4 B) 5,11,13 C) 7,24,26 D ) 8,15,17

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A right triangle is a triangle with one angle that is 90 degrees. In a right triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are called the legs. In this article, we will explore which of the given options forms a right triangle.

Understanding the Pythagorean Theorem

To determine if a triangle is a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, 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). This can be expressed as:

a^2 + b^2 = c^2

Option A: 2, 3, 4

Let's start by examining option A: 2, 3, 4. We can plug these values into the Pythagorean theorem to see if they form a right triangle.

  • a = 2
  • b = 3
  • c = 4

2^2 + 3^2 = 4^2 4 + 9 = 16 13 ≠ 16

Since 13 is not equal to 16, option A does not form a right triangle.

Option B: 5, 11, 13

Next, let's examine option B: 5, 11, 13. We can plug these values into the Pythagorean theorem to see if they form a right triangle.

  • a = 5
  • b = 11
  • c = 13

5^2 + 11^2 = 13^2 25 + 121 = 169 146 ≠ 169

Since 146 is not equal to 169, option B does not form a right triangle.

Option C: 7, 24, 26

Now, let's examine option C: 7, 24, 26. We can plug these values into the Pythagorean theorem to see if they form a right triangle.

  • a = 7
  • b = 24
  • c = 26

7^2 + 24^2 = 26^2 49 + 576 = 676 625 ≠ 676

Since 625 is not equal to 676, option C does not form a right triangle.

Option D: 8, 15, 17

Finally, let's examine option D: 8, 15, 17. We can plug these values into the Pythagorean theorem to see if they form a right triangle.

  • a = 8
  • b = 15
  • c = 17

8^2 + 15^2 = 17^2 64 + 225 = 289 289 = 289

Since 289 is equal to 289, option D forms a right triangle.

Conclusion

In conclusion, only option D: 8, 15, 17 forms a right triangle. The other options do not satisfy the Pythagorean theorem and therefore do not form a right triangle.

Why is the Pythagorean Theorem Important?

The Pythagorean theorem is a fundamental concept in geometry and trigonometry. It has numerous applications in various fields, including physics, engineering, and computer science. The theorem is used to calculate distances, heights, and angles in right triangles, which is essential in many real-world problems.

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 height of buildings, the length of shadows, and the distance between buildings.
  • Surveying: Surveyors use the Pythagorean theorem to calculate distances and angles between landmarks and reference points.
  • Physics: Physicists use the Pythagorean theorem to calculate distances, velocities, and accelerations in problems involving right triangles.
  • Computer Science: Computer scientists use the Pythagorean theorem to calculate distances and angles in computer graphics and game development.

Tips for Solving Right Triangle Problems

When solving right triangle problems, remember the following tips:

  • Use the Pythagorean theorem: The Pythagorean theorem is a powerful tool for solving right triangle problems.
  • Identify the hypotenuse: The hypotenuse is the side opposite the right angle.
  • Plug in values: Plug in the values of the sides into the Pythagorean theorem to solve for the unknown side.
  • Check your work: Check your work by plugging in the values into the Pythagorean theorem to ensure that the equation is satisfied.

Conclusion

Q: What is a right triangle?

A: A right triangle is a triangle with one angle that is 90 degrees. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a mathematical formula that describes the relationship between the lengths of the sides of a right triangle. It 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). This can be expressed as:

a^2 + b^2 = c^2

Q: How do I use the Pythagorean theorem to solve a right triangle problem?

A: To use the Pythagorean theorem to solve a right triangle problem, follow these steps:

  1. Identify the hypotenuse (c) and the other two sides (a and b).
  2. Plug in the values of the sides into the Pythagorean theorem.
  3. Solve for the unknown side.
  4. Check your work by plugging in the values into the Pythagorean theorem to ensure that the equation is satisfied.

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 identifying the hypotenuse: Make sure to identify the hypotenuse (c) and the other two sides (a and b) before plugging in values.
  • Not plugging in values correctly: Double-check that you have plugged in the correct values into the Pythagorean theorem.
  • Not solving for the correct side: Make sure to solve for the unknown side, not the hypotenuse or one of the legs.
  • Not checking your work: Always check your work by plugging in the values into the Pythagorean theorem to ensure that the equation is satisfied.

Q: Can I use the Pythagorean theorem to solve problems involving non-right triangles?

A: No, the Pythagorean theorem only applies to right triangles. If you are working with a non-right triangle, you will need to use a different formula or method to solve the problem.

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

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

  • Building design: Architects use the Pythagorean theorem to calculate the height of buildings, the length of shadows, and the distance between buildings.
  • Surveying: Surveyors use the Pythagorean theorem to calculate distances and angles between landmarks and reference points.
  • Physics: Physicists use the Pythagorean theorem to calculate distances, velocities, and accelerations in problems involving right triangles.
  • Computer science: Computer scientists use the Pythagorean theorem to calculate distances and angles in computer graphics and game development.

Q: How can I practice using the Pythagorean theorem?

A: There are many ways to practice using the Pythagorean theorem, including:

  • Working through practice problems: Try working through practice problems to get a feel for how to use the Pythagorean theorem.
  • Using online resources: There are many online resources available that provide practice problems and exercises to help you practice using the Pythagorean theorem.
  • Taking a math class: Consider taking a math class that covers the Pythagorean theorem and other geometric concepts.
  • Working with a tutor: If you are struggling with the Pythagorean theorem, consider working with a tutor who can provide one-on-one instruction and support.

Q: What are some common formulas and equations related to the Pythagorean theorem?

A: Some common formulas and equations related to the Pythagorean theorem include:

  • Pythagorean theorem: a^2 + b^2 = c^2
  • Distance formula: d = √(x2 - x1)^2 + (y2 - y1)^2
  • Midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2)
  • Slope formula: m = (y2 - y1)/(x2 - x1)

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

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

  • Not identifying the hypotenuse: Make sure to identify the hypotenuse (c) and the other two sides (a and b) before plugging in values.
  • Not plugging in values correctly: Double-check that you have plugged in the correct values into the Pythagorean theorem.
  • Not solving for the correct side: Make sure to solve for the unknown side, not the hypotenuse or one of the legs.
  • Not checking your work: Always check your work by plugging in the values into the Pythagorean theorem to ensure that the equation is satisfied.

Conclusion

In conclusion, the Pythagorean theorem is a fundamental concept in geometry and trigonometry that has numerous applications in various fields. By understanding the Pythagorean theorem and its applications, you can solve right triangle problems with ease. Remember to use the Pythagorean theorem, identify the hypotenuse, plug in values, and check your work to ensure that the equation is satisfied.