To determine the possible lengths of the third side of a triangle when two sides measure 5 inches and 12 inches, we can apply the triangle inequality theorem. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Given the sides:
- Side A = 5 inches
- Side B = 12 inches
- Side C = unknown (the third side)
According to the triangle inequality theorem, we have three conditions to satisfy:
- 1. The sum of the lengths of sides A and B must be greater than side C: 5 + 12 > C
- 2. The sum of the lengths of sides A and C must be greater than side B: 5 + C > 12
- 3. The sum of the lengths of sides B and C must be greater than side A: 12 + C > 5
Now, let’s solve each condition:
- From the first condition:
5 + 12 > C
This simplifies to C < 17. - From the second condition:
5 + C > 12
This simplifies to C > 12 – 5, or C > 7. - From the third condition:
12 + C > 5
This condition is always true since both 12 and C will always be positive, and thus C can be any positive value.
Considering the valid range from the first two conditions, we combine them:
7 < C < 17
Thus, the length of the third side must be greater than 7 inches and less than 17 inches. Therefore, the possible lengths for the third side of the triangle fall within the range of 8 inches to 16 inches, inclusive (since lengths must be whole numbers in this context).
In conclusion, the third side must be any length satisfying:
- 8 inches
- 9 inches
- 10 inches
- 11 inches
- 12 inches
- 13 inches
- 14 inches
- 15 inches
- 16 inches
So, all lengths from 8 to 16 inches, not including 7 and 17, are valid options for the length of the third side.