Determine whether these three side lengths will form a triangle: 6 cm, 8 cm, and 10 cm.(1 point)

Responses

They cannot form a triangle because the sum of any two of the side lengths is less than the length of the third side.
They cannot form a triangle because the sum of any two of the side lengths is less than the length of the third side.

They will form a triangle because the sum of any two of the side lengths is greater than the length of the third side.
They will form a triangle because the sum of any two of the side lengths is greater than the length of the third side.

They will form a triangle because the sum of any two of the side lengths is less than the length of the third side.
They will form a triangle because the sum of any two of the side lengths is less than the length of the third side.

They cannot form a triangle because the sum of any two of the side lengths is greater than the length of the third side.

1 answer

To determine if three side lengths can form a triangle, we can use the triangle inequality theorem. This theorem states that for any three sides \( a \), \( b \), and \( c \) (where the sides are ordered such that \( a \leq b \leq c \)), the following must hold:

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

For the side lengths given: 6 cm, 8 cm, and 10 cm:

  1. \( 6 + 8 = 14 \) which is greater than \( 10 \)
  2. \( 6 + 10 = 16 \) which is greater than \( 8 \)
  3. \( 8 + 10 = 18 \) which is greater than \( 6 \)

Since all three conditions are satisfied, the lengths 6 cm, 8 cm, and 10 cm can indeed form a triangle.

The correct response is: They will form a triangle because the sum of any two of the side lengths is greater than the length of the third side.