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 \

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Introduction to the Triangle Inequality Theorem

The Triangle Inequality Theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a triangle. This 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. In other words, the sum of the lengths of the two shorter sides must be greater than the length of the longest side. This theorem is essential in determining the range of possible values for the third side of a triangle.

Applying the Triangle Inequality Theorem to the Given Triangle

Given an acute triangle with sides measuring 10 cm and 16 cm, we need to determine the range of possible values for the third side. To do this, we can apply the Triangle Inequality Theorem. Let's denote the length of the third side as x.

Calculating the Range of Possible Values for the Third Side

According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, we have two possible combinations:

  • 10 + x > 16
  • 16 + x > 10

Solving the Inequalities

Let's solve the first inequality:

10 + x > 16

Subtracting 10 from both sides gives us:

x > 6

Now, let's solve the second inequality:

16 + x > 10

Subtracting 16 from both sides gives us:

x > -6

However, since the length of a side cannot be negative, we can ignore this inequality.

Determining the Upper Limit of the Third Side

To determine the upper limit of the third side, we need to consider the sum of the lengths of the two given sides. In this case, the sum of the lengths of the two given sides is 10 + 16 = 26. Therefore, the sum of the lengths of any two sides of the triangle must be less than 26.

Calculating the Upper Limit of the Third Side

Let's consider the combination 10 + x < 26. Subtracting 10 from both sides gives us:

x < 16

However, since the length of the third side must be greater than 6, we can conclude that the range of possible values for the third side is:

6 < x < 16

Conclusion

In conclusion, the range of possible values for the third side of the triangle is 6 < x < 16. This means that the length of the third side must be greater than 6 and less than 16.

Final Answer

The final answer is: 6<x<16\boxed{6 < x < 16}

Q: What is the Triangle Inequality Theorem?

A: The Triangle Inequality Theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a triangle. This 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.

Q: Why is the Triangle Inequality Theorem important?

A: The Triangle Inequality Theorem is essential in determining the range of possible values for the third side of a triangle. It helps us to understand the relationship between the lengths of the sides of a triangle and ensures that a triangle can be formed with the given side lengths.

Q: How do I apply the Triangle Inequality Theorem to a given triangle?

A: To apply the Triangle Inequality Theorem to a given triangle, you need to consider the sum of the lengths of any two sides of the triangle. You can then use this information to determine the range of possible values for the third side.

Q: What are the steps to solve a triangle inequality problem?

A: The steps to solve a triangle inequality problem are as follows:

  1. Write down the given side lengths of the triangle.
  2. Consider the sum of the lengths of any two sides of the triangle.
  3. Use the Triangle Inequality Theorem to determine the range of possible values for the third side.
  4. Solve the inequalities to find the upper and lower limits of the third side.

Q: Can a triangle be formed with side lengths that do not satisfy the Triangle Inequality Theorem?

A: No, a triangle cannot be formed with side lengths that do not satisfy the Triangle Inequality Theorem. The Triangle Inequality Theorem ensures that a triangle can be formed with the given side lengths.

Q: What is the significance of the upper limit of the third side?

A: The upper limit of the third side is the maximum possible value for the third side. It is determined by the sum of the lengths of the two given sides.

Q: Can the upper limit of the third side be negative?

A: No, the upper limit of the third side cannot be negative. The length of a side cannot be negative.

Q: How do I determine the lower limit of the third side?

A: The lower limit of the third side is the minimum possible value for the third side. It is determined by the difference between the sum of the lengths of the two given sides and the length of the longest side.

Q: Can the lower limit of the third side be negative?

A: No, the lower limit of the third side cannot be negative. The length of a side cannot be negative.

Q: What is the range of possible values for the third side?

A: The range of possible values for the third side is determined by the upper and lower limits of the third side. It is the set of all possible values for the third side that satisfy the Triangle Inequality Theorem.

Q: How do I find the range of possible values for the third side?

A: To find the range of possible values for the third side, you need to determine the upper and lower limits of the third side. You can then use this information to find the range of possible values for the third side.

Q: Can the range of possible values for the third side be empty?

A: Yes, the range of possible values for the third side can be empty. This occurs when the given side lengths do not satisfy the Triangle Inequality Theorem.

Q: What does it mean if the range of possible values for the third side is empty?

A: If the range of possible values for the third side is empty, it means that a triangle cannot be formed with the given side lengths. The given side lengths do not satisfy the Triangle Inequality Theorem.

Q: How do I determine if a triangle can be formed with given side lengths?

A: To determine if a triangle can be formed with given side lengths, you need to check if the given side lengths satisfy the Triangle Inequality Theorem. If they do, then a triangle can be formed with the given side lengths. If they do not, then a triangle cannot be formed with the given side lengths.

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:

  • Not considering the sum of the lengths of any two sides of the triangle.
  • Not solving the inequalities correctly.
  • Not checking if the given side lengths satisfy the Triangle Inequality Theorem.
  • Not considering the upper and lower limits of the third side.

Q: How do I practice applying the Triangle Inequality Theorem?

A: To practice applying the Triangle Inequality Theorem, you can try solving problems that involve determining the range of possible values for the third side of a triangle. You can also try creating your own problems and solving them using the Triangle Inequality Theorem.

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

A: Some real-world applications of the Triangle Inequality Theorem include:

  • Architecture: The Triangle Inequality Theorem is used to determine the stability of buildings and bridges.
  • Engineering: The Triangle Inequality Theorem is used to design and build structures such as bridges, tunnels, and buildings.
  • Physics: The Triangle Inequality Theorem is used to describe the motion of objects and the forces that act upon them.

Q: Can the Triangle Inequality Theorem be used to solve problems in other areas of mathematics?

A: Yes, the Triangle Inequality Theorem can be used to solve problems in other areas of mathematics such as algebra, geometry, and trigonometry.

Q: What are some common misconceptions about the Triangle Inequality Theorem?

A: Some common misconceptions about the Triangle Inequality Theorem include:

  • The Triangle Inequality Theorem only applies to triangles with right angles.
  • The Triangle Inequality Theorem only applies to triangles with equal side lengths.
  • The Triangle Inequality Theorem is only used to determine the range of possible values for the third side of a triangle.

Q: How do I overcome common misconceptions about the Triangle Inequality Theorem?

A: To overcome common misconceptions about the Triangle Inequality Theorem, you need to understand the theorem and its applications. You can also try solving problems that involve the Triangle Inequality Theorem and reading about its applications in different areas of mathematics.

Q: What are some tips for mastering the Triangle Inequality Theorem?

A: Some tips for mastering the Triangle Inequality Theorem include:

  • Practice solving problems that involve the Triangle Inequality Theorem.
  • Read about the applications of the Triangle Inequality Theorem in different areas of mathematics.
  • Try creating your own problems and solving them using the Triangle Inequality Theorem.
  • Join a study group or find a study partner to practice solving problems together.

Q: Can the Triangle Inequality Theorem be used to solve problems in other areas of science?

A: Yes, the Triangle Inequality Theorem can be used to solve problems in other areas of science such as physics, engineering, and computer science.

Q: What are some common applications of the Triangle Inequality Theorem in science?

A: Some common applications of the Triangle Inequality Theorem in science include:

  • Physics: The Triangle Inequality Theorem is used to describe the motion of objects and the forces that act upon them.
  • Engineering: The Triangle Inequality Theorem is used to design and build structures such as bridges, tunnels, and buildings.
  • Computer Science: The Triangle Inequality Theorem is used to develop algorithms and data structures.

Q: Can the Triangle Inequality Theorem be used to solve problems in other areas of technology?

A: Yes, the Triangle Inequality Theorem can be used to solve problems in other areas of technology such as computer science, engineering, and architecture.

Q: What are some common applications of the Triangle Inequality Theorem in technology?

A: Some common applications of the Triangle Inequality Theorem in technology include:

  • Computer Science: The Triangle Inequality Theorem is used to develop algorithms and data structures.
  • Engineering: The Triangle Inequality Theorem is used to design and build structures such as bridges, tunnels, and buildings.
  • Architecture: The Triangle Inequality Theorem is used to determine the stability of buildings and bridges.

Q: Can the Triangle Inequality Theorem be used to solve problems in other areas of business?

A: Yes, the Triangle Inequality Theorem can be used to solve problems in other areas of business such as finance, marketing, and management.

Q: What are some common applications of the Triangle Inequality Theorem in business?

A: Some common applications of the Triangle Inequality Theorem in business include:

  • Finance: The Triangle Inequality Theorem is used to determine the risk of investments and loans.
  • Marketing: The Triangle Inequality Theorem is used to determine the effectiveness of marketing campaigns.
  • Management: The Triangle Inequality Theorem is used to determine the efficiency of business operations.

Q: Can the Triangle Inequality Theorem be used to solve problems in other areas of education?

A: Yes, the Triangle Inequality Theorem can be used to solve problems in other areas of education such as mathematics, science, and engineering.

Q: What are some common applications of the Triangle Inequality Theorem in education?

A: Some common applications of the