Apply the Triangle Inequality Theorem to determine which three side lengths form a triangle.(1 point)

Question

Responses

10, 20, 15
10, 20, 15

8, 4, 12
8, 4, 12

20, 10, 30
20, 10, 30

8, 8, 20

Answer 1

The Triangle Inequality Theorem states that for any three sides of a triangle (a, b, c), the following conditions must be satisfied:

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Let's evaluate each set of side lengths:

1. 10, 20, 15:

   • \( 10 + 20 = 30 > 15 \) (True)
   • \( 10 + 15 = 25 > 20 \) (True)
   • \( 20 + 15 = 35 > 10 \) (True)
   • These can form a triangle.

2. 8, 4, 12:

   • \( 8 + 4 = 12 \) (Not greater than)
   • \( 8 + 12 = 20 > 4 \) (True)
   • \( 4 + 12 = 16 > 8 \) (True)
   • These cannot form a triangle.

3. 20, 10, 30:

   • \( 20 + 10 = 30 \) (Not greater than)
   • \( 20 + 30 = 50 > 10 \) (True)
   • \( 10 + 30 = 40 > 20 \) (True)
   • These cannot form a triangle.

4. 8, 8, 20:

   • \( 8 + 8 = 16 < 20 \) (Not greater than)
   • \( 8 + 20 = 28 > 8 \) (True)
   • \( 8 + 20 = 28 > 8 \) (True)
   • These cannot form a triangle.

The only set of sides that can form a triangle is 10, 20, 15.