Question

How many triangles exist with the given side lengths?



A.
Exactly one unique triangle exists with the given side lengths.

B.
No triangle exists with the given side lengths.

C.
More than one unique triangle exists with the given side lengths.

Answer 1

To determine how many triangles can be formed with the given side lengths, we need the specific lengths to analyze. However, I can describe a method for checking the possibilities:

1. **Triangle Inequality Theorem**: For three given lengths \( a \), \( b \), and \( c \), a triangle can exist if the following conditions are met:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)

2. **Unique Triangle Check**: Once you confirm that a triangle can exist using the triangle inequality, you can check if it is unique:
- If all three sides are of different lengths, only one triangle can be formed.
- If two sides are of equal length and one is different, also only one unique triangle exists (isosceles triangle).
- If all three sides are equal, exactly one unique triangle exists (equilateral triangle).
- If two sides are equal and the third side can vary based on certain constraints, there could be multiple triangles (but this typically applies to specific geometric conditions).

3. **Ambiguous Case (SSA)**: In cases where two sides and a non-included angle are given (Side-Side-Angle), there may be zero, one, or two triangles possible due to the ambiguous case.

With the specific side lengths provided, you can apply these principles to find the answer that fits the options given:

- **A**: Exactly one unique triangle exists with the given side lengths.
- **B**: No triangle exists with the given side lengths.
- **C**: More than one unique triangle exists with the given side lengths.

Feel free to provide the specific side lengths for a detailed assessment!