Apply the Triangle Inequality Theorem to determine whether sides with lengths of 4 inches, 8 inches, and 9 inches form a triangle.(1 point)

Question

Responses

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

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

No, all of the conditions are not met.
No, all of the conditions are not met.

No, because two out of three conditions are me

Answer 1

To determine if the lengths 4 inches, 8 inches, and 9 inches can form a triangle, we apply the Triangle Inequality Theorem. According to this theorem, for any three sides a, b, and c, the following conditions must hold:

1. a + b > c
2. a + c > b
3. b + c > a

Let's assign the lengths:

• a = 4
• b = 8
• c = 9

Now we check each condition:

1. 4 + 8 > 9:

   • 12 > 9, which is true.

2. 4 + 9 > 8:

   • 13 > 8, which is true.

3. 8 + 9 > 4:

   • 17 > 4, which is true.

All three conditions are met. Therefore, the sides with lengths of 4 inches, 8 inches, and 9 inches can indeed form a triangle.

The correct response is:

Yes, all of the conditions are met.

Answer 2

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

Answer 3

To determine whether the lengths 4.24 inches, 9.89 inches, and 8.75 inches can form a triangle, we will apply the Triangle Inequality Theorem.

Let’s denote the sides as:

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

Now we check the three conditions:

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

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

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

   • \( 4.24 + 8.75 > 9.89 \)
   • \( 13.99 > 9.89 \) (True)

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

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

All three conditions of the Triangle Inequality Theorem are satisfied:

• \( a + b > c \)
• \( a + c > b \)
• \( b + c > a \)

So, the three sides can form a triangle.

The correct response 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 a triangle can be formed from sides of lengths 7 millimeters, 8 millimeters, and 9 millimeters.(1 point)
Responses

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

Yes, because the sum of any two sides is greater than the third

Answer 5

To determine whether a triangle can be formed from sides of lengths 7 millimeters, 8 millimeters, and 9 millimeters, we will use the Triangle Inequality Theorem.

Let’s denote the sides as:

• \( a = 7 \) mm
• \( b = 8 \) mm
• \( c = 9 \) mm

Now we check the three conditions:

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

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

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

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

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

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

All three conditions of the Triangle Inequality Theorem are satisfied:

• \( a + b > c \)
• \( a + c > b \)
• \( b + c > a \)

Therefore, the sides can indeed form a triangle.

The correct response is:

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

Answer 6

i meant side

Answer 7

Thank you for the clarification! To determine whether a triangle can be formed from sides of lengths 7 millimeters, 8 millimeters, and 9 millimeters, let's use the Triangle Inequality Theorem again, focusing on the fact that we are considering "sides."

Let’s denote the sides as:

• \( a = 7 \) mm
• \( b = 8 \) mm
• \( c = 9 \) mm

Now we check the three conditions of the Triangle Inequality Theorem:

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

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

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

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

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

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

All three conditions are satisfied.

Thus, it is indeed possible to form a triangle with sides of lengths 7 millimeters, 8 millimeters, and 9 millimeters.

So, the correct response is:

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