Apply the triangle inequality theorem to determine which three side lengths form a triangle
10,20,15
8,8,20
8,4,12
20,10,30
To determine which three side lengths form a triangle, we apply the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
1. For the first set of side lengths (10, 20, 15):
- 10 + 15 = 25 > 20 (so 10 and 15 can form a triangle)
- 10 + 20 = 30 > 15 (so 10 and 20 can form a triangle)
- 20 + 15 = 35 > 10 (so 15 and 20 can form a triangle)
Therefore, all three side lengths (10, 20, 15) can form a triangle.
2. For the second set of side lengths (8, 8, 20):
- 8 + 8 = 16 < 20 (8 and 8 cannot form a triangle)
- 8 + 20 = 28 > 8 (so 8 and 20 can form a triangle)
- 8 + 20 = 28 > 8 (so 8 and 20 can form a triangle)
Therefore, only two side lengths (8, 20) can form a triangle.
3. For the third set of side lengths (8, 4, 12):
- 4 + 8 = 12 < 12 (4 and 8 cannot form a triangle)
- 4 + 12 = 16 < 8 (4 and 12 cannot form a triangle)
- 8 + 12 = 20 > 4 (so 8 and 12 can form a triangle)
Therefore, only two side lengths (8, 12) can form a triangle.
4. For the fourth set of side lengths (20, 10, 30):
- 10 + 20 = 30 > 30 (so 10 and 20 can form a triangle)
- 10 + 30 = 40 > 20 (so 10 and 30 can form a triangle)
- 20 + 30 = 50 > 10 (so 20 and 30 can form a triangle)
Therefore, all three side lengths (20, 10, 30) can form a triangle.