Select The Correct Answer.A Triangle Has One Side Of Length 29 Units And Another Of Length 40 Units. Determine The Range In Which The Length Of The Third Side Must Lie.A. $ -11 \ \textless \ X \ \textless \ 69 $ B. $ 11 \leq X \leq 69

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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 in which the length of the third side of a triangle must lie, given the lengths of the other two sides.

Applying the Triangle Inequality Theorem

In this problem, we are given a triangle with one side of length 29 units and another of length 40 units. We need to determine the range in which the length of the third side must lie. To do this, we will apply the triangle inequality theorem.

Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Given values:

  • Side 1: 29 units
  • Side 2: 40 units

Unknown value: Length of the third side (x)

Applying the theorem:

  • Side 1 + Side 2 > x

  • 29 + 40 > x

  • 69 > x

  • Side 1 + x > Side 2

  • 29 + x > 40

  • x > 11

  • Side 2 + x > Side 1

  • 40 + x > 29

  • x > -11

Determining the range:

From the above inequalities, we can see that the length of the third side (x) must be greater than -11 and less than 69. Therefore, the range in which the length of the third side must lie is:

-11 < x < 69

Conclusion: The correct answer is A. −11 \textless x \textless 69-11 \ \textless \ x \ \textless \ 69.

Discussion: The triangle inequality theorem is a fundamental concept in geometry that has numerous applications in various fields, including physics, engineering, and computer science. Understanding this theorem is essential in solving problems related to triangles and their properties.

Real-world applications:

  • In physics, the triangle inequality theorem is used to calculate the distance between two points in a coordinate system.
  • In engineering, the theorem is used to design and analyze structures, such as bridges and buildings.
  • In computer science, the theorem is used in algorithms for solving problems related to geometry and spatial reasoning.

Common mistakes:

  • Many students make the mistake of assuming that the length of the third side must be equal to the sum of the lengths of the other two sides.
  • Others may assume that the length of the third side must be less than the sum of the lengths of the other two sides.

Tips and tricks:

  • When applying the triangle inequality theorem, it is essential to consider all three inequalities: Side 1 + Side 2 > x, Side 1 + x > Side 2, and Side 2 + x > Side 1.
  • To avoid common mistakes, it is crucial to carefully read and understand the problem statement and the theorem.

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 third side.

Q: Why is the triangle inequality theorem important?

A: The triangle inequality theorem is essential in determining the range in which the length of the third side of a triangle must lie, given the lengths of the other two sides. It also has numerous applications in various fields, including physics, engineering, and computer science.

Q: How do I apply the triangle inequality theorem?

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

  • Side 1 + Side 2 > x
  • Side 1 + x > Side 2
  • Side 2 + x > Side 1

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:

  • Assuming that the length of the third side must be equal to the sum of the lengths of the other two sides.
  • Assuming that the length of the third side must be less than the sum of the lengths of the other two sides.

Q: How do I determine the range in which the length of the third side must lie?

A: To determine the range in which the length of the third side must lie, you need to consider the three inequalities mentioned above. By solving these inequalities, you can determine the minimum and maximum values of the length of the third side.

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

A: Some real-world applications of the triangle inequality theorem include:

  • Calculating the distance between two points in a coordinate system.
  • Designing and analyzing structures, such as bridges and buildings.
  • Solving problems related to geometry and spatial reasoning in computer science.

Q: How do I use the triangle inequality theorem to solve problems?

A: To use the triangle inequality theorem to solve problems, you need to:

  1. Read and understand the problem statement.
  2. Identify the lengths of the sides of the triangle.
  3. Apply the triangle inequality theorem by considering the three inequalities.
  4. Solve the inequalities to determine the range in which the length of the third side must lie.

Q: What are some tips and tricks for applying the triangle inequality theorem?

A: Some tips and tricks for applying the triangle inequality theorem include:

  • Carefully reading and understanding the problem statement and the theorem.
  • Considering all three inequalities when applying the theorem.
  • Avoiding common mistakes, such as assuming that the length of the third side must be equal to the sum of the lengths of the other two sides.

Conclusion: In conclusion, the triangle inequality theorem is a fundamental concept in geometry that has numerous applications in various fields. By understanding the theorem and applying it correctly, you can solve problems related to triangles and their properties. Remember to carefully read and understand the problem statement, consider all three inequalities, and avoid common mistakes.