To determine which set of side lengths does not form a triangle, we can apply the triangle inequality theorem. This theorem states that for any triangle with side lengths \( a \), \( b \), and \( c \), the following conditions must hold:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let's examine each given set of lengths:
-
(18, 10, 8):
- \( 18 + 10 > 8 \) (True)
- \( 18 + 8 > 10 \) (True)
- \( 10 + 8 > 18 \) (False) → Cannot form a triangle
-
(18, 10, 8) (Repeating):
- As above, it cannot form a triangle.
-
(18, 11, 8):
- \( 18 + 11 > 8 \) (True)
- \( 18 + 8 > 11 \) (True)
- \( 11 + 8 > 18 \) (False) → Cannot form a triangle
-
(9, 17, 11):
- \( 9 + 17 > 11 \) (True)
- \( 9 + 11 > 17 \) (False) → Cannot form a triangle
-
(19, 9, 11):
- \( 19 + 9 > 11 \) (True)
- \( 19 + 11 > 9 \) (True)
- \( 9 + 11 > 19 \) (False) → Cannot form a triangle
From the analysis above, the sets that do not form a triangle are:
- (18, 10, 8)
- (18, 11, 8)
- (9, 17, 11)
- (19, 9, 11)
Therefore, there are multiple sets that do not form a triangle. If only one set is to be chosen, any of those above could suffice.
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