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

Responses

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 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 cannot form a triangle because the sum of any two of the side lengths is greater 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.

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.
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1 answer

To determine if three side lengths can form a triangle, we can use the triangle inequality theorem, which states that for any triangle with sides of lengths \( a \), \( b \), and \( c \):

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

Let's apply this to the given side lengths: 6 cm, 8 cm, and 10 cm.

  1. Check \( 6 + 8 > 10 \):

    • \( 14 > 10 \) (True)
  2. Check \( 6 + 10 > 8 \):

    • \( 16 > 8 \) (True)
  3. Check \( 8 + 10 > 6 \):

    • \( 18 > 6 \) (True)

Since all three conditions of the triangle inequality theorem are satisfied, these side lengths can indeed form a triangle.

Thus, 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.