To find the range of possible lengths for the third side of a triangle when two sides are known, we can use the triangle inequality theorem. The 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.
Given two sides \( a = 4 \) cm and \( b = 17 \) cm, we can denote the third side as \( c \). According to the triangle inequality, we have the following conditions:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Now let's apply these conditions:
-
From \( a + b > c \):
\( 4 + 17 > c \)
\( 21 > c \)
Therefore, \( c < 21 \). -
From \( a + c > b \):
\( 4 + c > 17 \)
\( c > 17 - 4 \)
\( c > 13 \). -
From \( b + c > a \):
\( 17 + c > 4 \)
This condition is always satisfied since both \( c \) and \( 17 \) are positive lengths.
Combining these results, we have:
- \( c > 13 \)
- \( c < 21 \)
Thus, the range of possible lengths for the third side \( c \) is:
\[ \boxed{(13 \text{ cm}, 21 \text{ cm})} \]
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