A Triangle Has Side Lengths Measuring $20 \, \text{cm}$, $5 \, \text{cm}$, And $n \, \text{cm}$. Which Describes The Possible Values Of $n$?A. $15 \ \textless \ N \ \textless \ 20$B. $15 \ \textless

Introduction

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 crucial in determining the possible values of the third side of a triangle, given the lengths of the other two sides. In this article, we will explore the Triangle Inequality Theorem and apply it to a specific problem involving a triangle with side lengths measuring $20 \, \text{cm}$, $5 \, \text{cm}$, and $n \, \text{cm}$.

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

Mathematically, this can be expressed as:

• $a + b > c$
• $a + c > b$
• $b + c > a$
• $|a - b| < c$
• $|a - c| < b$
• $|b - c| < a$

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

Applying the Triangle Inequality Theorem to the Problem

Given the side lengths of the triangle as $20 \, \text{cm}$, $5 \, \text{cm}$, and $n \, \text{cm}$, we can apply the Triangle Inequality Theorem to determine the possible values of $n$.

First, we need to consider the sum of the lengths of any two sides of the triangle. In this case, we have:

• $20 + 5 > n$
• $20 + n > 5$
• $5 + n > 20$

Simplifying these inequalities, we get:

• $25 > n$
• $n > -15$
• $n > 15$

Next, we need to consider the difference between the lengths of any two sides of the triangle. In this case, we have:

• $|20 - 5| < n$
• $|20 - n| < 5$
• $|5 - n| < 20$

Simplifying these inequalities, we get:

• $15 < n$
• $n < 25$
• $n < 25$

Determining the Possible Values of n

From the inequalities derived above, we can see that the possible values of $n$ are:

• $n > 15$
• $n < 25$

However, we also need to consider the fact that the length of the third side of a triangle cannot be equal to the sum of the lengths of the other two sides. Therefore, we need to exclude the values of $n$ that are equal to $25$.

Conclusion

In conclusion, the possible values of $n$ are $15 < n < 25$. This means that the length of the third side of the triangle must be greater than $15$ and less than $25$.

Final Answer

The final answer is: $\boxed{15 < n < 25}$

Introduction

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. In our previous article, we explored the Triangle Inequality Theorem and applied it to a specific problem involving a triangle with side lengths measuring $20 \, \text{cm}$, $5 \, \text{cm}$, and $n \, \text{cm}$. In this article, we will answer some frequently asked questions about the Triangle Inequality Theorem.

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: What are the three inequalities of the Triangle Inequality Theorem?

A: The three inequalities of the Triangle Inequality Theorem are:

• $a + b > c$
• $a + c > b$
• $b + c > a$

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

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

A: To apply the Triangle Inequality Theorem to a problem, you need to consider the sum of the lengths of any two sides of the triangle and the difference between the lengths of any two sides of the triangle. You can then use the inequalities of the Triangle Inequality Theorem to determine the possible values of the third side of the triangle.

Q: What happens if the sum of the lengths of two sides of a triangle is equal to the length of the third side?

A: If the sum of the lengths of two sides of a triangle is equal to the length of the third side, then the triangle is degenerate, meaning that it has zero area.

Q: Can the Triangle Inequality Theorem be applied to all types of triangles?

A: Yes, the Triangle Inequality Theorem can be applied to all types of triangles, including acute triangles, right triangles, and obtuse triangles.

Q: How do I determine the possible values of the third side of a triangle?

A: To determine the possible values of the third side of a triangle, you need to consider the inequalities of the Triangle Inequality Theorem and the fact that the length of the third side of a triangle cannot be equal to the sum of the lengths of the other two sides.

Q: What is the significance of the Triangle Inequality Theorem in real-life applications?

A: The Triangle Inequality Theorem has many real-life applications, including navigation, surveying, and engineering. It is used to determine the possible values of distances and angles in various problems.

Conclusion

In conclusion, the Triangle Inequality Theorem is a fundamental concept in geometry that has many real-life applications. By understanding the Triangle Inequality Theorem, you can solve problems involving triangles and determine the possible values of the third side of a triangle.

Final Answer

The final answer is: 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.