Question

Apply the Triangle Inequality Theorem to determine whether sides with lengths of 4.24 inches, 9.89 inches, and 8.75 inches form a triangle.(1 point) Responses No, because the sum of any two sides is greater than the third side. No, because the sum of any two sides is greater than the third side. Yes, because the sum of any two sides is greater than the third side. Yes, because the sum of any two sides is greater than the third side. No, because the sum of any two sides is less than the third side. No, because the sum of any two sides is less than the third side. Yes, because the sum of any two sides is less than the third side.

Answer 1

To determine whether the sides with lengths of 4.24 inches, 9.89 inches, and 8.75 inches can form a triangle, we will apply the Triangle Inequality Theorem. This theorem states that for three sides \(a\), \(b\), and \(c\) to form a triangle, the following conditions must all be true:

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

Let's assign the sides as follows:

• \(a = 4.24\)
• \(b = 9.89\)
• \(c = 8.75\)

Now, we will check each condition:

1. Check if \(a + b > c\):

   • \(4.24 + 9.89 = 14.13\)
   • \(14.13 > 8.75\) (True)

2. Check if \(a + c > b\):

   • \(4.24 + 8.75 = 13.00\)
   • \(13.00 > 9.89\) (True)

3. Check if \(b + c > a\):

   • \(9.89 + 8.75 = 18.64\)
   • \(18.64 > 4.24\) (True)

Since all three conditions are satisfied, the sides can indeed form a triangle.

The correct answer is: Yes, because the sum of any two sides is greater than the third side.

Answer 2

Apply the Triangle Inequality Theorem to determine whether a triangle can be formed from sides of lengths 7 millimeters, 8 millimeters, and 9 millimeters.(1 point) Responses Yes, because the sum of any two sides is greater than the third side. Yes, because the sum of any two sides is greater than the third side. No, because the sum of any two sides is greater than the third side. No, because the sum of any two sides is greater than the third side. No, because the sum of any two sides is less than the third side. No, because the sum of any two sides is less than the third side. Yes, because the sum of any two sides is less than the third side.

Answer 3

To determine whether the sides with lengths 7 millimeters, 8 millimeters, and 9 millimeters can form a triangle, we will apply the Triangle Inequality Theorem.

As a reminder, the Triangle Inequality Theorem states that for three sides \(a\), \(b\), and \(c\) to form a triangle, the following three conditions must be met:

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

Let's assign the sides as follows:

• \(a = 7\)
• \(b = 8\)
• \(c = 9\)

Now we will check each of the conditions:

1. Check if \(a + b > c\):

   • \(7 + 8 = 15\)
   • \(15 > 9\) (True)

2. Check if \(a + c > b\):

   • \(7 + 9 = 16\)
   • \(16 > 8\) (True)

3. Check if \(b + c > a\):

   • \(8 + 9 = 17\)
   • \(17 > 7\) (True)

Since all three conditions are satisfied, the sides can indeed form a triangle.

The correct answer is: Yes, because the sum of any two sides is greater than the third side.

Answer 4

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.

Answer 5

To determine whether the sides with lengths of 5 inches, 10 inches, and 15 inches can form a triangle, we will apply the Triangle Inequality Theorem. This theorem requires that the following conditions be satisfied:

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

Let's assign the side lengths as follows:

• \(a = 5\) inches
• \(b = 10\) inches
• \(c = 15\) inches

Now we will check each condition:

1. Check if \(a + b > c\):

   • \(5 + 10 = 15\)
   • \(15 > 15\) (False)

2. Check if \(a + c > b\):

   • \(5 + 15 = 20\)
   • \(20 > 10\) (True)

3. Check if \(b + c > a\):

   • \(10 + 15 = 25\)
   • \(25 > 5\) (True)

The first condition \(5 + 10 > 15\) is not satisfied, as it is equal to 15, not greater. Therefore, a triangle cannot be formed with these side lengths.

The correct response is: No, because the sum of 5 + 10 is not greater than 15.

Answer 6

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 No, because 3 + 4 is less than 9. No, because 3 plus 4 is less than 9. Yes, because the sum of any two sides is less than the third side. Yes, because the sum of any two sides is less than the third side. Yes, the majority of the conditions are met. Yes, the majority of the conditions are met. Yes, because 3 + 4 is less than 9.

Answer 7

To determine whether the sides with lengths of 3 inches, 4 inches, and 9 inches can form a triangle, we will again use the Triangle Inequality Theorem, which requires that the following conditions be met:

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

Let's assign the side lengths as follows:

• \(a = 3\) inches
• \(b = 4\) inches
• \(c = 9\) inches

Now we will check each condition:

1. Check if \(a + b > c\):

   • \(3 + 4 = 7\)
   • \(7 > 9\) (False)

2. Check if \(a + c > b\):

   • \(3 + 9 = 12\)
   • \(12 > 4\) (True)

3. Check if \(b + c > a\):

   • \(4 + 9 = 13\)
   • \(13 > 3\) (True)

The first condition \(3 + 4 > 9\) is not satisfied, as it is less than 9. Because this condition fails, the three lengths cannot form a triangle.

The correct response is: No, because 3 + 4 is less than 9.