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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 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 side. Yes, because the sum of any two sides is greater than the third side. Skip to navigation page 8 of 8

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

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

Let's denote the sides as follows:

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

Now we will check each of the three inequalities:

  1. \(a + b > c\)

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

  2. \(a + c > b\)

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

  3. \(b + c > a\)

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

Since all three inequalities are true, the lengths can indeed form a triangle. Hence, the correct response would be:

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

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 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. 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 less than the third side.

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

Let:

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

We need to check the following inequalities:

  1. \(a + b > c\): \[ 7 + 8 = 15 > 9 \quad (\text{True}) \]

  2. \(a + c > b\): \[ 7 + 9 = 16 > 8 \quad (\text{True}) \]

  3. \(b + c > a\): \[ 8 + 9 = 17 > 7 \quad (\text{True}) \]

Since all three inequalities are satisfied, the lengths can indeed form a triangle.

Therefore, the correct response is:

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