The Triangle Inequality Theorem states that for any three sides \( a, b, c \) of a triangle, the following inequalities must hold:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let's evaluate each set of side lengths given:
-
8, 4, 12
- \( 8 + 4 = 12 \) (not greater than)
- \( 8 + 12 = 20 > 4 \)
- \( 4 + 12 = 16 > 8 \)
- Does not form a triangle.
-
8, 8, 20
- \( 8 + 8 = 16 < 20 \) (not greater than)
- \( 8 + 20 = 28 > 8 \)
- \( 8 + 20 = 28 > 8 \)
- Does not form a triangle.
-
10, 20, 15
- \( 10 + 20 = 30 > 15 \)
- \( 10 + 15 = 25 > 20 \)
- \( 20 + 15 = 35 > 10 \)
- Forms a triangle.
-
20, 10, 30
- \( 20 + 10 = 30 \) (not greater than)
- \( 20 + 30 = 50 > 10 \)
- \( 10 + 30 = 40 > 20 \)
- Does not form a triangle.
Based on this analysis, the only set of side lengths that forms a triangle is 10, 20, 15.
1 answer