A Triangle Has Side Lengths Of 5 Cm And 8 Cm. The Third Side Has A Length Of X X X . Which Inequality Describes All The Possible Lengths In Centimeters For Side X X X ?A. 5 ≤ X ≤ 8 5 \leq X \leq 8 5 ≤ X ≤ 8 B. $5 \ \textless \ X \ \textless \
Understanding 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 third side. This theorem is essential in determining the possible lengths of the third side of a triangle when the lengths of the other two sides are known.
Applying the Triangle Inequality Theorem to the Given Problem
In this problem, we are given a triangle with side lengths of 5 cm and 8 cm. We need to find the possible lengths of the third side, denoted as . To do this, we will apply the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Finding the Lower Bound of the Third Side
The lower bound of the third side can be found by subtracting the length of the shortest side from the length of the longest side. In this case, the shortest side is 5 cm, and the longest side is 8 cm. Therefore, the lower bound of the third side is:
This means that the third side must be greater than 3 cm.
Finding the Upper Bound of the Third Side
The upper bound of the third side can be found by adding the lengths of the other two sides. In this case, the lengths of the other two sides are 5 cm and 8 cm. Therefore, the upper bound of the third side is:
This means that the third side must be less than 13 cm.
Writing the Inequality
Based on the lower and upper bounds found above, we can write the inequality that describes all the possible lengths of the third side:
Conclusion
In conclusion, the triangle inequality theorem is a powerful tool for determining the possible lengths of the third side of a triangle when the lengths of the other two sides are known. By applying this theorem, we can find the lower and upper bounds of the third side and write the inequality that describes all the possible lengths.
Answer
The correct answer is B. .
Understanding 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 third side. This theorem is essential in determining the possible lengths of the third side of a triangle when the lengths of the other two sides are known.
Q&A
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: How do I apply the triangle inequality theorem to find the possible lengths of the third side?
A: To apply the triangle inequality theorem, you need to find the lower and upper bounds of the third side. The lower bound can be found by subtracting the length of the shortest side from the length of the longest side, while the upper bound can be found by adding the lengths of the other two sides.
Q: What is the lower bound of the third side?
A: The lower bound of the third side can be found by subtracting the length of the shortest side from the length of the longest side. For example, if the lengths of the other two sides are 5 cm and 8 cm, the lower bound of the third side would be 8 - 5 = 3 cm.
Q: What is the upper bound of the third side?
A: The upper bound of the third side can be found by adding the lengths of the other two sides. For example, if the lengths of the other two sides are 5 cm and 8 cm, the upper bound of the third side would be 5 + 8 = 13 cm.
Q: How do I write the inequality that describes all the possible lengths of the third side?
A: To write the inequality, you need to use the lower and upper bounds found above. For example, if the lower bound is 3 cm and the upper bound is 13 cm, the inequality would be 3 < x < 13.
Q: What is the correct answer to the problem: A triangle has side lengths of 5 cm and 8 cm. The third side has a length of . Which inequality describes all the possible lengths in centimeters for side ?
A: The correct answer is B. .
Conclusion
In conclusion, the triangle inequality theorem is a powerful tool for determining the possible lengths of the third side of a triangle when the lengths of the other two sides are known. By applying this theorem, you can find the lower and upper bounds of the third side and write the inequality that describes all the possible lengths.
Frequently Asked Questions
Q: What is the triangle inequality theorem used for?
A: The triangle inequality theorem is used to determine the possible lengths of the third side of a triangle when the lengths of the other two sides are known.
Q: How do I use the triangle inequality theorem in real-life situations?
A: The triangle inequality theorem can be used in real-life situations such as architecture, engineering, and design. For example, when designing a building, you need to ensure that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Q: Can the triangle inequality theorem be used for non-geometric shapes?
A: No, the triangle inequality theorem can only be used for geometric shapes such as triangles.
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 finding the lower and upper bounds of the third side
- Not writing the inequality correctly
- Not considering the signs of the lengths of the sides
Conclusion
In conclusion, the triangle inequality theorem is a powerful tool for determining the possible lengths of the third side of a triangle when the lengths of the other two sides are known. By applying this theorem, you can find the lower and upper bounds of the third side and write the inequality that describes all the possible lengths.