An Acute Triangle Has Sides Measuring 10 Cm And 16 Cm. The Length Of The Third Side Is Unknown. Which Best Describes The Range Of Possible Values For The Third Side Of The Triangle?A. $x \ \textless \ 12.5, X \ \textgreater \ 18.9$ B.

by ADMIN 240 views

The triangle inequality theorem is a fundamental concept in geometry that states the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This theorem is essential in determining the range of possible values for the third side of a triangle when the lengths of the other two sides are known.

Applying the Triangle Inequality Theorem

In the given problem, we have an acute triangle with sides measuring 10 cm and 16 cm. The length of the third side is unknown. To find the range of possible values for the third side, we can apply the triangle inequality theorem.

According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Let's denote the length of the third side as x. Then, we have the following inequalities:

  • 10 + 16 > x
  • 10 + x > 16
  • 16 + x > 10

Simplifying these inequalities, we get:

  • 26 > x
  • x > 6
  • x > -6

However, since the length of a side cannot be negative, we can ignore the last inequality. Therefore, the range of possible values for the third side is x > 6 and x < 26.

Comparing the Options

Now, let's compare the range of possible values for the third side (x > 6 and x < 26) with the given options:

A. x \textless 12.5,x \textgreater 18.9x \ \textless \ 12.5, x \ \textgreater \ 18.9

B. x \textgreater 6,x \textless 26x \ \textgreater \ 6, x \ \textless \ 26

Based on the calculations, option B is the correct answer, as it accurately represents the range of possible values for the third side of the triangle.

Conclusion

In conclusion, the triangle inequality theorem is a powerful tool for determining the range of possible values for the third side of a triangle when the lengths of the other two sides are known. By applying this theorem, we can find the correct range of values for the third side, which is essential in various mathematical and real-world applications.

Key Takeaways

  • The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
  • The range of possible values for the third side can be determined by applying the triangle inequality theorem.
  • The correct range of values for the third side is x > 6 and x < 26.

Further Reading

For more information on the triangle inequality theorem and its applications, refer to the following resources:

The triangle inequality theorem is a fundamental concept in geometry that has numerous applications in mathematics and real-world problems. However, it can be challenging to understand and apply, especially for beginners. In this article, we will address some frequently asked questions (FAQs) about the triangle inequality theorem to help you better comprehend this concept.

Q: What is the triangle inequality theorem?

A: The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This theorem is essential in determining the range of possible values for the third side of a triangle when the lengths of the other two sides are known.

Q: How do I apply the triangle inequality theorem?

A: To apply the triangle inequality theorem, you need to follow these steps:

  1. Identify the lengths of the two known sides of the triangle.
  2. Add the lengths of the two known sides to get the sum.
  3. Compare the sum with the length of the unknown side.
  4. If the sum is greater than the length of the unknown side, then the unknown side is valid.
  5. If the sum is less than or equal to the length of the unknown side, then the unknown side is not valid.

Q: What are the limitations of the triangle inequality theorem?

A: The triangle inequality theorem has several limitations:

  • It only applies to triangles with positive side lengths.
  • It does not provide information about the shape or size of the triangle.
  • It only determines the range of possible values for the third side, but not the exact value.

Q: Can I use the triangle inequality theorem to find the length of the third side?

A: Yes, you can use the triangle inequality theorem to find the range of possible values for the third side. However, it may not provide the exact length of the third side. To find the exact length, you need to use other mathematical techniques, such as the Pythagorean theorem or Heron's formula.

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

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

  • Architecture: To determine the feasibility of building designs and ensure that the structure is stable.
  • Engineering: To design and optimize systems, such as bridges and buildings.
  • Computer Science: To develop algorithms and data structures, such as graph theory and network analysis.
  • Physics: To model and analyze physical systems, such as motion and forces.

Q: Can I use the triangle inequality theorem to solve problems involving obtuse or right triangles?

A: Yes, you can use the triangle inequality theorem to solve problems involving obtuse or right triangles. However, you need to be careful when applying the theorem, as the inequalities may change depending on the type of triangle.

Q: What are some common mistakes to avoid when applying the triangle inequality theorem?

A: Some common mistakes to avoid when applying the triangle inequality theorem include:

  • Failing to consider the signs of the side lengths.
  • Ignoring the inequalities when the triangle is obtuse or right.
  • Not checking for the validity of the unknown side.

Conclusion

The triangle inequality theorem is a powerful tool for determining the range of possible values for the third side of a triangle when the lengths of the other two sides are known. By understanding the theorem and its applications, you can develop a deeper appreciation for the beauty and power of mathematics. Remember to be careful when applying the theorem, and avoid common mistakes to ensure accurate results.

Key Takeaways

  • The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
  • The theorem has numerous real-world applications, including architecture, engineering, computer science, and physics.
  • Be careful when applying the theorem, and avoid common mistakes to ensure accurate results.

Further Reading

For more information on the triangle inequality theorem and its applications, refer to the following resources:

By understanding the triangle inequality theorem and its applications, you can develop a deeper appreciation for the beauty and power of mathematics.