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Answer :
To determine which measurement is not a possibility for the third side of a triangle with sides measuring 23 meters and 49 meters, we need to apply the Triangle Inequality Theorem. This 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.
Here's how we can check each option:
1. Calculate the Range for the Third Side:
- The third side must be less than the sum of the other two sides:
[tex]\[
\text{Sum of the two sides} = 23 + 49 = 72 \text{ meters}
\][/tex]
- The third side must be greater than the absolute difference of the other two sides:
[tex]\[
\text{Absolute difference of the two sides} = |23 - 49| = 26 \text{ meters}
\][/tex]
2. Determine Possible Third Sides:
- The third side must be greater than 26 meters and less than 72 meters. Thus, it must satisfy:
[tex]\[
26 < \text{Third side} < 72
\][/tex]
3. Evaluate Each Given Option:
- 27 meters: Since [tex]\(26 < 27 < 72\)[/tex], this is possible.
- 45 meters: Since [tex]\(26 < 45 < 72\)[/tex], this is possible.
- 36 meters: Since [tex]\(26 < 36 < 72\)[/tex], this is possible.
- 71 meters: Since [tex]\(26 < 71 < 72\)[/tex], this is possible.
- 25 meters: Since [tex]\(25\)[/tex] is not greater than 26, this is not possible.
- 32 meters: Since [tex]\(26 < 32 < 72\)[/tex], this is possible.
Based on the calculations, the measurement that is not a possible length for the third side of the triangle is 25 meters.
Here's how we can check each option:
1. Calculate the Range for the Third Side:
- The third side must be less than the sum of the other two sides:
[tex]\[
\text{Sum of the two sides} = 23 + 49 = 72 \text{ meters}
\][/tex]
- The third side must be greater than the absolute difference of the other two sides:
[tex]\[
\text{Absolute difference of the two sides} = |23 - 49| = 26 \text{ meters}
\][/tex]
2. Determine Possible Third Sides:
- The third side must be greater than 26 meters and less than 72 meters. Thus, it must satisfy:
[tex]\[
26 < \text{Third side} < 72
\][/tex]
3. Evaluate Each Given Option:
- 27 meters: Since [tex]\(26 < 27 < 72\)[/tex], this is possible.
- 45 meters: Since [tex]\(26 < 45 < 72\)[/tex], this is possible.
- 36 meters: Since [tex]\(26 < 36 < 72\)[/tex], this is possible.
- 71 meters: Since [tex]\(26 < 71 < 72\)[/tex], this is possible.
- 25 meters: Since [tex]\(25\)[/tex] is not greater than 26, this is not possible.
- 32 meters: Since [tex]\(26 < 32 < 72\)[/tex], this is possible.
Based on the calculations, the measurement that is not a possible length for the third side of the triangle is 25 meters.
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