A Triangle Has The Following Side Lengths: 6 Cm, 6 Cm. What Is The Smallest The Third Side Length Could Be To Still Make A Triangle? Enter Your Answer As A Number.______ Cm

Feb 27, 2025 by ADMIN 173 views

**Triangle Inequality Theorem: Finding the Smallest Third Side Length**

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 essential in determining the validity of a triangle and is used extensively in various mathematical and real-world applications. In this article, we will explore the triangle inequality theorem and use it to find the smallest possible length of the third side of a triangle with given side lengths of 6 cm and 6 cm.

Understanding the 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 represented as:

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

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

Applying the Triangle Inequality Theorem

Given the side lengths of 6 cm and 6 cm, we can use the triangle inequality theorem to find the smallest possible length of the third side. Let's assume the third side length is x.

Using the first inequality, we get:

6 + 6 > x 12 > x

This means that the third side length must be less than 12 cm.

Using the second inequality, we get:

6 + x > 6 x > 0

This means that the third side length must be greater than 0 cm.

Using the third inequality, we get:

6 + x > 6 x > 0

This is the same as the second inequality.

Finding the Smallest Third Side Length

To find the smallest possible length of the third side, we need to find the smallest value of x that satisfies the inequalities.

From the first inequality, we know that x must be less than 12 cm.

From the second inequality, we know that x must be greater than 0 cm.

The smallest possible value of x that satisfies both inequalities is 0 cm. However, this is not possible since a triangle cannot have a side length of 0 cm.

The next smallest possible value of x that satisfies both inequalities is 1 cm. However, this is still not possible since the sum of the lengths of the two given sides (6 cm + 6 cm) is 12 cm, and the third side length must be less than 12 cm.

The next smallest possible value of x that satisfies both inequalities is 2 cm. However, this is still not possible since the sum of the lengths of the two given sides (6 cm + 6 cm) is 12 cm, and the third side length must be less than 12 cm.

We can continue this process until we find the smallest possible value of x that satisfies both inequalities.

Conclusion

Using the triangle inequality theorem, we can find the smallest possible length of the third side of a triangle with given side lengths of 6 cm and 6 cm. The smallest possible value of x that satisfies both inequalities is 2 cm. However, this is still not possible since the sum of the lengths of the two given sides (6 cm + 6 cm) is 12 cm, and the third side length must be less than 12 cm.

The next smallest possible value of x that satisfies both inequalities is 0 cm. However, this is not possible since a triangle cannot have a side length of 0 cm.

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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 essential in determining the validity of a triangle and is used extensively in various mathematical and real-world applications. In this article, we will explore the triangle inequality theorem and answer some frequently asked questions related to it.

Q: What is the triangle inequality theorem?

A: The triangle inequality 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 important because it helps us determine the validity of a triangle. If the sum of the lengths of any two sides of a triangle is not greater than the length of the third side, then the triangle is not valid.

Q: How do I apply the triangle inequality theorem?

A: To apply the triangle inequality theorem, you need to follow these steps:

Write down the lengths of the three sides of the triangle.
Add the lengths of any two sides together.
Check if the sum is greater than the length of the third side.
If the sum is greater than the length of the third side, then the triangle is valid. If not, then the triangle is not valid.

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

A: If the sum of the lengths of any two sides of a triangle is equal to the length of the third side, then the triangle is not valid. This is because the sum of the lengths of any two sides of a triangle must be greater than the length of the third side, not equal to it.

Q: Can a triangle have a side length of 0 cm?

A: No, a triangle cannot have a side length of 0 cm. This is because the sum of the lengths of any two sides of a triangle must be greater than the length of the third side, and 0 cm is not greater than any positive number.

Q: Can a triangle have two sides of equal length?

A: Yes, a triangle can have two sides of equal length. In fact, it is possible for a triangle to have all three sides of equal length.

Q: How do I find the smallest possible length of the third side of a triangle?

A: To find the smallest possible length of the third side of a triangle, you need to follow these steps:

Write down the lengths of the two given sides of the triangle.
Add the lengths of the two given sides together.
Subtract the length of the third side from the sum.
The result is the smallest possible length of the third side.

Q: What is the smallest possible length of the third side of a triangle with given side lengths of 6 cm and 6 cm?

A: To find the smallest possible length of the third side of a triangle with given side lengths of 6 cm and 6 cm, we need to follow the steps outlined above.

Write down the lengths of the two given sides of the triangle: 6 cm and 6 cm.
Add the lengths of the two given sides together: 6 cm + 6 cm = 12 cm.
Subtract the length of the third side from the sum: 12 cm - x, where x is the length of the third side.
The result is the smallest possible length of the third side: 12 cm - x.

Since the sum of the lengths of the two given sides (12 cm) is greater than the length of the third side (x), we know that x must be less than 12 cm.

The smallest possible value of x that satisfies this inequality is 0 cm. However, this is not possible since a triangle cannot have a side length of 0 cm.

The next smallest possible value of x that satisfies this inequality is 1 cm. However, this is still not possible since the sum of the lengths of the two given sides (12 cm) is greater than the length of the third side (1 cm).

We can continue this process until we find the smallest possible value of x that satisfies this inequality.

Conclusion

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 validity of a triangle and is used extensively in various mathematical and real-world applications. In this article, we have explored the triangle inequality theorem and answered some frequently asked questions related to it.

The smallest possible length of the third side of a triangle with given side lengths of 6 cm and 6 cm is 0 cm. However, this is not possible since a triangle cannot have a side length of 0 cm.

We can continue this process until we find the smallest possible value of x that satisfies this inequality.

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