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Apply the Triangle Inequality Theorem to determine whether sides with lengths of 5 inches, 10 inches, and 15 inches form a triangle.(1 point)

Responses

Yes, the majority of the conditions are met.
Yes, the majority of the conditions are met.

Yes, because the sum of 5+10 is not greater than 15.
Yes, because the sum of 5 plus 10 is not greater than 15.

Yes, because the sum of 5+10 is less than or equal to 15.
Yes, because the sum of 5 plus 10 is less than or equal to 15.

No, because the sum of 5+10 is not greater than 15.

3 answers

To determine whether the sides with lengths of 5 inches, 10 inches, and 15 inches can form a triangle, we can use the Triangle Inequality Theorem, which states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following must be true:

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

Let's apply this to the given sides:

  • Let \( a = 5 \), \( b = 10 \), and \( c = 15 \).

Now we check the inequalities:

  1. Check \( 5 + 10 > 15 \):

    • \( 15 > 15 \) (which is false)

  2. Check \( 5 + 15 > 10 \):

    • \( 20 > 10 \) (which is true)

  3. Check \( 10 + 15 > 5 \):

    • \( 25 > 5 \) (which is true)

Since the first condition is not satisfied (the sum of the two smaller sides is not greater than the length of the largest side), the sides do not meet the conditions of the Triangle Inequality Theorem.

Therefore, the correct response is:

No, because the sum of 5+10 is not greater than 15.

Apply the Triangle Inequality Theorem to determine whether sides with lengths of 3 inches, 4 inches, and 9 inches form a triangle.(1 point)
Responses

Yes, the majority of the conditions are met.
Yes, the majority of the conditions are met.

No, because 3+4 is less than 9.
No, because 3 plus 4 is less than 9.

Yes, because 3+4 is less than 9.
Yes, because 3 plus 4 is less than 9.

Yes, because the sum of any two sides is less than the third side.

To determine whether the sides with lengths of 3 inches, 4 inches, and 9 inches can form a triangle, we will use the Triangle Inequality Theorem.

According to the theorem, the following conditions must hold true for three lengths \( a \), \( b \), and \( c \) to form a triangle:

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

For the sides given:

  • Let \( a = 3 \), \( b = 4 \), and \( c = 9 \).

Now, let's check the inequalities:

  1. Check \( 3 + 4 > 9 \):

    • \( 7 > 9 \) (which is false)

  2. Check \( 3 + 9 > 4 \):

    • \( 12 > 4 \) (which is true)

  3. Check \( 4 + 9 > 3 \):

    • \( 13 > 3 \) (which is true)

Since the first condition is not satisfied (the sum of the two shorter sides is not greater than the length of the longest side), these sides cannot form a triangle.

Thus, the correct response is:

No, because 3+4 is less than 9.