To determine if the given side lengths of 6 inches, 8 inches, and 10 inches can form a triangle, we can apply the triangle inequality theorem. This theorem states that for any triangle with side lengths \( a \), \( b \), and \( c \):
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let's check each condition with the given side lengths:
-
\( 6 + 8 > 10 \)
- \( 14 > 10 \) (True)
-
\( 6 + 10 > 8 \)
- \( 16 > 8 \) (True)
-
\( 8 + 10 > 6 \)
- \( 18 > 6 \) (True)
Since all three conditions are satisfied, the side lengths comply with the triangle inequality theorem, which means they can form a triangle.
Next, we will check if the triangle is unique. The given side lengths (6, 8, and 10) form a right triangle, as they satisfy the Pythagorean theorem \( a^2 + b^2 = c^2 \):
- \( 6^2 + 8^2 = 10^2 \)
- \( 36 + 64 = 100 \)
- \( 100 = 100 \) (True)
Since the side lengths define a unique configuration of a triangle (specifically, a right triangle), we conclude that there is a unique triangle with these side lengths.
Therefore, the answer is:
unique triangle
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