Question

Tell whether it is possible to make a triangle with the given side lengths. 1,1,1

Short answer

Yes, a triangle can be formed with side lengths 1, 1, and 1.

Step-by-step solution

- Understand Triangle Inequality Theorem

The Triangle Inequality Theorem states that for any three side lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

- Apply the Theorem to the Given Side Lengths

Evaluate the sum of each pair of side lengths to see if it is greater than the third side: 1. Check if 1 + 1 > 1 2. Check if 1 + 1 > 1 3. Check if 1 + 1 > 1

- Verify Each Condition

Each of the following inequalities must hold true:1. 1 + 1 = 2, which is greater than 1.2. 1 + 1 = 2, which is greater than 1.3. 1 + 1 = 2, which is greater than 1.

- Conclusion

Since all three conditions are satisfied, the given side lengths can form a triangle.

Key concepts

Triangle Formation

Forming a triangle depends on the relationship between its side lengths. Specifically, we use the Triangle Inequality Theorem. This theorem helps us determine if three side lengths can form a triangle. The theorem states: For any three side lengths, the sum of the lengths of any two sides must be greater than the length of the third side.
In simpler terms, if you take any two sides of a potential triangle and add them, their sum must be more than the third side.
This ensures that the sides can 'connect' to form a closed shape, which is essential for a proper triangle.

Side Lengths

When we talk about side lengths in triangle formation, we mean the numerical values that represent the lengths of each side. Consider the side lengths 1, 1, and 1, as given in the exercise. To see if these lengths can form a triangle, we use the Triangle Inequality Theorem and do the following calculations:

• Check if 1 + 1 > 1
• Check if 1 + 1 > 1
• Check if 1 + 1 > 1

Each of these conditions must be true. In our case:

• 1 + 1 equals 2, which is greater than 1
• 1 + 1 equals 2, which is greater than 1
• 1 + 1 equals 2, which is greater than 1

Since all conditions are satisfied, these side lengths can indeed form a triangle.

Geometry

Geometry is the branch of mathematics that deals with shapes, sizes, and the properties of space. Understanding the basics of geometric principles, like the Triangle Inequality Theorem, is crucial for solving problems related to triangle formation.
In geometry, a triangle is one of the simplest forms, consisting of three sides and three angles. The concept of side lengths and their relationships as dictated by the Triangle Inequality Theorem helps ensure the stability and validity of the triangle.
By grasping these fundamental rules, you can solve more complex geometric problems and understand the structure and properties of various shapes more clearly. Remember, the principles of geometry are building blocks for many other areas of mathematics and science.