Question

Tell whether it is possible to make a triangle with the given side lengths. 25,25,200

Short answer

No, it is not possible.

Step-by-step solution

- Understand the Triangle Inequality Theorem

To determine if it is possible to form a triangle with three given side lengths, use the Triangle Inequality Theorem. This theorem states that for any three side lengths, the sum of any two side lengths must be greater than the third side length.

- Check the first condition

Add the first two sides and ensure this sum is greater than the third side: \[ 25 + 25 > 200 \]Calculate the sum: \[ 50 > 200 \]This condition is false.

- Check the second condition

Add the first and third sides and ensure this sum is greater than the second side: \[ 25 + 200 > 25 \]Calculate the sum: \[ 225 > 25 \]This condition is true.

- Check the third condition

Add the second and third sides and ensure this sum is greater than the first side: \[ 25 + 200 > 25 \]Calculate the sum: \[ 225 > 25 \]This condition is true.

- Conclusion

At least one condition from the Triangle Inequality Theorem is false. Therefore, it is not possible to form a triangle with the given side lengths.

Key concepts

triangle side lengths

To determine if three given side lengths can form a triangle, you need to understand that the side lengths interact based on specific rules. These rules ensure that the sides can physically connect to form a closed shape, in this case, a triangle. Imagine having three sticks; for them to connect end-to-end and form a triangle, each pair of sides must be sufficiently long to reach the other side's endpoint. This brings us to an important geometric rule called the Triangle Inequality Theorem.

triangle conditions

The Triangle Inequality Theorem provides crucial conditions that must be met for three side lengths to actually form a triangle. The theorem states:

• The sum of the lengths of any two sides must be greater than the length of the third side.

If this condition isn't met for any combination of two sides out of the three given lengths, then a triangle cannot be formed. For example, let's consider checking if a triangle can be formed with side lengths of 25, 25, and 200. Applying the Triangle Inequality Theorem involves three checks:

• First side length + Second side length > Third side length ewline 25 + 25 > 200 ewline 50 > 200 (This is false)
• First side length + Third side length > Second side length ewline 25 + 200 > 25 ewline 225 > 25 (This is true)
• Second side length + Third side length > First side length ewline 25 + 200 > 25 ewline 225 > 25 (This is true)

Since the first check is false, it is not possible to form a triangle with the given side lengths.

inequality theorem

The Triangle Inequality Theorem is essential for understanding why certain side lengths cannot make a triangle. It's not just a random rule but a fundamental principle in geometry. Ensure that the sum of any two sides is always greater than the third side. This concept helps in:

Validating triangle formations in geometry problems.

Ensuring structural integrity in architecture and various engineering fields.

Understanding properties of different types of triangles, such as equilateral, isosceles, and scalene.

As we saw in the given problem, although two conditions were satisfied, the first condition was not. This single failure makes the formation of a triangle impossible. Hence, the Triangle Inequality Theorem is a reliable check to verify the feasibility of creating a triangle with specific side lengths.