An Acute Triangle Has Two Sides Measuring 8 Cm And 10 Cm. What Is The Best Representation Of The Possible Values For The Third Side, $s$?A. $2 \ \textless \ S \ \textless \ 18$B. $6 \ \textless \ S \ \textless \

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 remaining side. This theorem is crucial in determining the possible values for the third side of a triangle when two sides are given. In this article, we will explore the triangle inequality theorem and its application to find the possible values for the third side of a triangle.

The Triangle Inequality Theorem

The triangle inequality theorem can be stated as follows:

• The sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
• The difference between the lengths of any two sides of a triangle must be less than the length of the remaining side.

Mathematically, this can be represented as:

• $a + b > c$
• $|a - b| < c$

where $a$, $b$, and $c$ are the lengths of the sides of the triangle.

Applying the Triangle Inequality Theorem

Given that an acute triangle has two sides measuring 8 cm and 10 cm, we can use the triangle inequality theorem to find the possible values for the third side, $s$. Let's apply the theorem to the given sides:

• $a = 8$ cm
• $b = 10$ cm
• $c = s$

Using the first part of the theorem, we get:

• $a + b > c$
• $8 + 10 > s$
• $18 > s$

Using the second part of the theorem, we get:

• $|a - b| < c$
• $|8 - 10| < s$
• $2 < s$

Therefore, the possible values for the third side, $s$, are given by the inequality:

• $2 < s < 18$

Conclusion

In conclusion, the triangle inequality theorem is a powerful tool for determining the possible values for the third side of a triangle when two sides are given. By applying the theorem to the given sides of the acute triangle, we found that the possible values for the third side, $s$, are given by the inequality $2 < s < 18$. This result is consistent with the options provided in the discussion category.

Final Answer

The final answer is:

• $2 < s < 18$

This answer is consistent with the options provided in the discussion category, and it represents the possible values for the third side, $s$, of the acute triangle.

References

• Triangle Inequality Theorem
• Geometry

Additional Resources

• Triangle Inequality Theorem
• Geometry

Related Topics

• Triangle Inequality Theorem
• Geometry
• Acute Triangle

FAQs

• What is the triangle inequality theorem?
  • 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 remaining side.
• How do I apply the triangle inequality theorem?
  • To apply the triangle inequality theorem, you need to use the following inequalities:
    • $a + b > c$
    • $|a - b| < c$
  • where $a$, $b$, and $c$ are the lengths of the sides of the triangle.
• What are the possible values for the third side, $s$, of an acute triangle?
  • The possible values for the third side, $s$, of an acute triangle are given by the inequality $2 < s < 18$.
    Q&A: Understanding the Triangle Inequality Theorem =====================================================

In our previous article, we explored the triangle inequality theorem and its application to find the possible values for the third side of a triangle. In this article, we will continue to delve deeper into the triangle inequality theorem and answer some of the most frequently asked questions related to this concept.

Q: What is the triangle inequality theorem?

A: 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 remaining side. This theorem is crucial in determining the possible values for the third side of a triangle when two sides are given.

Q: How do I apply the triangle inequality theorem?

A: To apply the triangle inequality theorem, you need to use the following inequalities:

• $a + b > c$
• $|a - b| < c$

where $a$, $b$, and $c$ are the lengths of the sides of the triangle.

Q: What are the possible values for the third side, $s$, of an acute triangle?

A: The possible values for the third side, $s$, of an acute triangle are given by the inequality $2 < s < 18$. This result is consistent with the options provided in the discussion category.

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

A: Yes, you can use the triangle inequality theorem to find the length of the third side of a triangle. By applying the theorem to the given sides, you can determine the possible values for the third side.

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

A: The triangle inequality theorem has several limitations. For example, it does not provide information about the exact length of the third side, but rather the possible range of values. Additionally, the theorem assumes that the triangle is a valid triangle, meaning that the sum of the lengths of any two sides must be greater than the length of the remaining side.

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

A: Yes, you can use the triangle inequality theorem to find the length of the third side of a right triangle. However, you need to take into account the fact that the triangle is a right triangle, meaning that one of the angles is 90 degrees.

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

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

• Geometry: The theorem is used to determine the possible values for the third side of a triangle.
• Trigonometry: The theorem is used to find the length of the third side of a triangle.
• Physics: The theorem is used to determine the possible values for the length of a spring.
• Engineering: The theorem is used to determine the possible values for the length of a beam.

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

A: No, you cannot use the triangle inequality theorem to find the length of the third side of a triangle with negative side lengths. The theorem assumes that the side lengths are positive.

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

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

• Not considering the sign of the side lengths: The theorem assumes that the side lengths are positive.
• Not considering the fact that the triangle is a valid triangle: The theorem assumes that the sum of the lengths of any two sides must be greater than the length of the remaining side.
• Not considering the fact that the triangle is a right triangle: The theorem assumes that the triangle is not a right triangle.

Conclusion

In conclusion, 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 remaining side. By understanding the theorem and its application, you can determine the possible values for the third side of a triangle. We hope that this article has provided you with a better understanding of the triangle inequality theorem and its applications.