Math 8 Quarter 4-Week 2 Applying Theorems On Triangle Inequalities Acute Angels TVmpH-VLC Media Player Media Playback Audio Video Subtitle Tools View Help Activity 1. Two Sides Of A Triangle Have The Measures 7 And 11. Find The Range Of Possible

Mar 11, 2025 by ADMIN 246 views

**Math 8 Quarter 4-Week 2: Applying Theorems on Triangle Inequalities**

Introduction

In the world of geometry, theorems play a crucial role in understanding and solving problems related to triangles. One such theorem is 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. In this article, we will delve into the application of this theorem, specifically focusing on finding the range of possible values for the third side of a triangle when two sides have measures 7 and 11.

Understanding the Triangle Inequality Theorem

The Triangle Inequality Theorem is a fundamental concept in geometry that helps us determine the validity of a triangle based on the lengths of its sides. The theorem states that for any triangle with sides of lengths a, b, and c, the following inequalities must hold:

a + b > c
a + c > b
b + c > a

These inequalities ensure that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. This theorem is essential in understanding the properties of triangles and is widely used in various mathematical applications.

Applying the Theorem to Find the Range of Possible Values

Now, let's apply the Triangle Inequality Theorem to find the range of possible values for the third side of a triangle when two sides have measures 7 and 11. We can represent the third side as x.

Using the first inequality, we have:

7 + 11 > x 18 > x

This means that the third side must be less than 18.

Using the second inequality, we have:

7 + x > 11 x > 4

This means that the third side must be greater than 4.

Using the third inequality, we have:

11 + x > 7 x > -4

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

Finding the Range of Possible Values

Combining the inequalities from the previous section, we can conclude that the range of possible values for the third side of the triangle is:

4 < x < 18

This means that the third side of the triangle must be greater than 4 and less than 18.

Visualizing the Solution

To better understand the solution, let's visualize the range of possible values for the third side of the triangle. We can represent the range as a number line, with the lower bound at 4 and the upper bound at 18.

| | 4 | 5 | 6 | ... | 17 | 18 |

The third side of the triangle can take any value within this range, as long as it satisfies the Triangle Inequality Theorem.

Conclusion

In this article, we applied the Triangle Inequality Theorem to find the range of possible values for the third side of a triangle when two sides have measures 7 and 11. We used the theorem to derive three inequalities, which helped us determine the lower and upper bounds of the range. The final answer is a range of possible values, rather than a single value, which highlights the importance of considering multiple possibilities in mathematical problems.

Activity 1: Exploring the Triangle Inequality Theorem

Try the following activity to explore the Triangle Inequality Theorem further:

Draw a triangle with sides of lengths 7, 11, and x.
Use the Triangle Inequality Theorem to determine the range of possible values for the third side of the triangle.
Visualize the solution using a number line.
Experiment with different values of x to see how the range of possible values changes.

Media Playback

To enhance your understanding of the Triangle Inequality Theorem, watch the following video:

Triangle Inequality Theorem

This video provides a visual explanation of the theorem and its application in solving problems related to triangles.

Subtitle Tools

To access the subtitle tools, follow these steps:

Click on the "CC" button in the video player.
Select the language of your choice from the dropdown menu.
Adjust the subtitle settings as needed.

View Help

For more information on the Triangle Inequality Theorem, refer to the following resources:

Introduction

In our previous article, we explored the application of the Triangle Inequality Theorem in finding the range of possible values for the third side of a triangle when two sides have measures 7 and 11. In this article, we will address some common questions and concerns related to the Triangle Inequality Theorem and its application in solving problems related to triangles.

Q&A

Q: What is the Triangle Inequality Theorem?

A: The Triangle Inequality Theorem is a fundamental concept in geometry that 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 understanding the properties of triangles and is widely used in various mathematical applications, including solving problems related to triangles, determining the validity of a triangle, and finding the range of possible values for the third side of a triangle.

Q: How do I apply the Triangle Inequality Theorem to find the range of possible values for the third side of a triangle?

A: To apply the Triangle Inequality Theorem, you need to use the following inequalities:

a + b > c
a + c > b
b + c > a

where a, b, and c are the lengths of the sides of the triangle. By solving these inequalities, you can determine the range of possible values for the third side of the triangle.

Q: What if the inequalities do not have a solution?

A: If the inequalities do not have a solution, it means that the triangle is not valid, and the given side lengths do not form a triangle.

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. However, you need to have the lengths of the other two sides of the triangle to apply the theorem.

Q: Are there any exceptions to the Triangle Inequality Theorem?

A: No, there are no exceptions to the Triangle Inequality Theorem. The theorem is a fundamental concept in geometry that applies to all triangles.

Q: Can I use the Triangle Inequality Theorem to solve problems related to right triangles?

A: Yes, you can use the Triangle Inequality Theorem to solve problems related to right triangles. However, you need to consider the properties of right triangles, such as the Pythagorean theorem, to find the length of the third side of the triangle.

Q: Are there any online resources that can help me understand the Triangle Inequality Theorem?

A: Yes, there are many online resources that can help you understand the Triangle Inequality Theorem, including videos, tutorials, and practice problems. Some popular resources include: