Question

iliana claims that she can construct a triangle from a 12 - inch rod, a 36 - inch rod, and a 39.4 - inch rod. which statement explains whether she is correct?
o iliana cannot construct a triangle because the sum of the two smaller sides will not be greater than the third side.
o iliana cannot construct a triangle because the sum of the two smaller sides will be greater than the third side.
o iliana will be able to construct a triangle because the sum of any two sides is greater than the third side.
o iliana will be able to construct a triangle because the sum of any two sides is less than the third side.

Explanation:

To determine if a triangle can be formed, we use the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the third side. Let's check the given side lengths: 12 inches, 36 inches, and 39.4 inches.

• First, sum the two smaller sides: \(12 + 36 = 48\) inches.
• Now, compare this sum to the largest side (39.4 inches). Since \(48>39.4\), the sum of the two smaller sides is greater than the third side. Also, we can check the other combinations: \(12 + 39.4 = 51.4>36\) and \(36 + 39.4 = 75.4>12\). So, the sum of any two sides is greater than the third side, which means a triangle can be constructed.

Now let's analyze the options:

• Option 1: Incorrect, because the sum of the two smaller sides (\(12 + 36 = 48\)) is greater than the third side (39.4).
• Option 2: Incorrect, the reasoning is wrong. If the sum of two smaller sides is greater than the third side, a triangle can be formed, and this option says she cannot construct a triangle for that reason.
• Option 3: Correct, as we verified that the sum of any two sides is greater than the third side, so a triangle can be constructed.
• Option 4: Incorrect, the triangle inequality theorem states the sum of any two sides must be greater than the third side, not less.

Answer:

C. Iliana will be able to construct a triangle because the sum of any two sides is greater than the third side.