To find the range of possible lengths for the third side of a triangle when two sides are known, you can use 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.
Let the lengths of the two known sides be \(a = 9\) yards and \(b = 32\) yards, and let \(c\) be the length of the third side. According to the triangle inequality, we have the following three inequalities:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Now, substituting the values \(a = 9\) and \(b = 32\) into these inequalities:
-
\(9 + 32 > c\)
- \(41 > c\)
- Therefore, \(c < 41\)
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\(9 + c > 32\)
- \(c > 32 - 9\)
- \(c > 23\)
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\(32 + c > 9\) (This inequality will always hold true since \(c\) must be positive.)
Combining the results from the first two inequalities, we get:
\[ 23 < c < 41 \]
Thus, the range of possible lengths for the third side \(c\) is:
\[ c \text{ must be greater than } 23 \text{ yards and less than } 41 \text{ yards.} \]
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