Chapter 1: Problem 19
Tell whether it is possible to make a triangle with the given side lengths. 3,4,5
Short Answer
Expert verified
Yes, 3, 4, and 5 can form a triangle.
Step by step solution
01
- Understand the Triangle Inequality Theorem
To determine if three side lengths can form a triangle, the sum of any two side lengths must be greater than the third side length. Mathematically, for side lengths a, b, and c: 1. a + b > c2. a + c > b3. b + c > a
02
- Check the first condition
Check if the sum of the first two side lengths is greater than the third side length. For sides 3, 4, and 5: 3 + 4 > 5 Since 7 > 5, this condition is true.
03
- Check the second condition
Check if the sum of the first and third side lengths is greater than the second side length:3 + 5 > 4 Since 8 > 4, this condition is true.
04
- Check the third condition
Check if the sum of the second and third side lengths is greater than the first side length:4 + 5 > 3Since 9 > 3, this condition is true.
05
- Conclusion
Since all three conditions of the Triangle Inequality Theorem are satisfied, the side lengths 3, 4, and 5 can form a triangle.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
triangle formation
To understand if three given side lengths can form a triangle, we need to consider the Triangle Inequality Theorem. This theorem provides a set of rules that the side lengths must follow to form a valid triangle. Essentially, it ensures that the side lengths fit together properly to enclose a space. Using the example from our exercise, we can explore how these rules work.
Given the side lengths 3, 4, and 5, we need to verify three conditions:
* The sum of the first two sides must be greater than the third: 3 + 4 > 5. This is true because 7 > 5.
* The sum of the first and third sides must be greater than the second: 3 + 5 > 4. This is true because 8 > 4.
* The sum of the second and third sides must be greater than the first: 4 + 5 > 3. This is true because 9 > 3.
Since all these conditions are met, the side lengths can form a triangle. This is an easy but crucial step in determining if a set of side lengths is viable in geometry.
Given the side lengths 3, 4, and 5, we need to verify three conditions:
* The sum of the first two sides must be greater than the third: 3 + 4 > 5. This is true because 7 > 5.
* The sum of the first and third sides must be greater than the second: 3 + 5 > 4. This is true because 8 > 4.
* The sum of the second and third sides must be greater than the first: 4 + 5 > 3. This is true because 9 > 3.
Since all these conditions are met, the side lengths can form a triangle. This is an easy but crucial step in determining if a set of side lengths is viable in geometry.
geometry
Geometry is the branch of mathematics that deals with shapes, sizes, and the properties of space. It involves understanding the relationships and properties of points, lines, surfaces, and solids. One of the most essential shapes studied in geometry is the triangle.
Triangles are unique because their properties are straightforward, yet they form the basis for more complex geometric concepts. The exercise provided asks whether three side lengths can create a triangle, hinging on the Triangle Inequality Theorem. This theorem helps us understand fundamental geometric relationships.
In our example, we have side lengths 3, 4, and 5. By using the theorem, we confirmed these side lengths can form a triangle. This exercise is fundamental in geometry because it applies a basic yet powerful rule. Understanding how to validate the side lengths with this theorem reveals the importance of logical thinking and reasoning in geometry. The concepts we use here become the foundation for more complex geometrical analyses as we progress in mathematical studies.
Triangles are unique because their properties are straightforward, yet they form the basis for more complex geometric concepts. The exercise provided asks whether three side lengths can create a triangle, hinging on the Triangle Inequality Theorem. This theorem helps us understand fundamental geometric relationships.
In our example, we have side lengths 3, 4, and 5. By using the theorem, we confirmed these side lengths can form a triangle. This exercise is fundamental in geometry because it applies a basic yet powerful rule. Understanding how to validate the side lengths with this theorem reveals the importance of logical thinking and reasoning in geometry. The concepts we use here become the foundation for more complex geometrical analyses as we progress in mathematical studies.
side lengths
Side lengths are the measurable extents of a triangle's edges. In geometry, the relationship between the side lengths plays a critical role in determining the type and possibility of forming a triangle.
For instance, the given side lengths 3, 4, and 5 in our exercise are specific values that we need to check against the Triangle Inequality Theorem. When we validated these by checking:
* 3 + 4 > 5
* 3 + 5 > 4
* 4 + 5 > 3
We established that these side lengths satisfy the theorem. Generally, side lengths determine not only whether a triangle can form but also its type.
When all sides are equal, we have an equilateral triangle. When two sides are equal, it's an isosceles triangle. If all sides are different, like in our example, it's a scalene triangle. Each type has unique properties and implications for further geometrical studies. Understanding these basics about side lengths leads to a deeper comprehension of triangle properties and classification in geometry.
For instance, the given side lengths 3, 4, and 5 in our exercise are specific values that we need to check against the Triangle Inequality Theorem. When we validated these by checking:
* 3 + 4 > 5
* 3 + 5 > 4
* 4 + 5 > 3
We established that these side lengths satisfy the theorem. Generally, side lengths determine not only whether a triangle can form but also its type.
When all sides are equal, we have an equilateral triangle. When two sides are equal, it's an isosceles triangle. If all sides are different, like in our example, it's a scalene triangle. Each type has unique properties and implications for further geometrical studies. Understanding these basics about side lengths leads to a deeper comprehension of triangle properties and classification in geometry.
