Apply the Triangle Inequality Theorem

Answer 1

The Triangle Inequality Theorem states that, for any triangle with sides of lengths \( a \), \( b \), and \( c \):

1. The sum of the lengths of any two sides must be greater than the length of the third side. This can be expressed with the following inequalities:
   • \( a + b > c \)
   • \( a + c > b \)
   • \( b + c > a \)

To apply the Triangle Inequality Theorem, you can:

1. Check if Three Lengths Can Form a Triangle: Given any three lengths, say \( a = 3 \), \( b = 4 \), and \( c = 5 \):

   • Check \( a + b > c \): \( 3 + 4 > 5 \) (True)
   • Check \( a + c > b \): \( 3 + 5 > 4 \) (True)
   • Check \( b + c > a \): \( 4 + 5 > 3 \) (True)

   Since all three conditions are satisfied, the lengths 3, 4, and 5 can form a triangle.

2. Determine if a Set of Lengths Can Form a Triangle: Given lengths 7, 10, and 15:

   • Check \( 7 + 10 > 15 \): \( 17 > 15 \) (True)
   • Check \( 7 + 15 > 10 \): \( 22 > 10 \) (True)
   • Check \( 10 + 15 > 7 \): \( 25 > 7 \) (True)

   All conditions hold, so these lengths can form a triangle.

3. Real-World Applications: In real-world scenarios, you might use the Triangle Inequality to ensure that certain cable lengths can connect points without exceeding lengths between connection points.

If you have specific lengths in mind that you'd like to check using the Triangle Inequality Theorem, please provide them!