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?

iliana cannot construct a triangle because the sum of the two smaller sides will not be greater than the third side.
iliana cannot construct a triangle because the sum of the two smaller sides will be greater than the third side.
iliana will be able to construct a triangle because the sum of any two sides is greater than the third side.
iliana will be able to construct a triangle because the sum of any two sides is less than the third side.

Explanation:

Identify the given side lengths

The three given side lengths are:

$$ a = 12,\quad b = 36,\quad c = 39.4 $$

Apply the Triangle Inequality Theorem

To form a triangle, the sum of the lengths of any two sides must be strictly greater than the length of the remaining side. We check the sum of the two smaller sides:

$$ a + b = 12 + 36 = 48 $$

Compare the sum to the longest side

We compare this sum to the longest side, \(c = 39.4\):

$$ 48 > 39.4 $$

Since the sum of the two smaller sides is greater than the third side, a triangle can be constructed.

Match with the correct statement

Since \(12 + 36 > 39.4\), Iliana is correct and can construct the triangle. The statement that correctly explains this is: "Iliana will be able to construct a triangle because the sum of any two sides is greater than the third side."

Answer:

• Iliana cannot construct a triangle because the sum of the two smaller sides will not be greater than the third side.
• Iliana cannot construct a triangle because the sum of the two smaller sides will be greater than the third side.
• Iliana will be able to construct a triangle because the sum of any two sides is greater than the third side. (Correct answer)
• Iliana will be able to construct a triangle because the sum of any two sides is less than the third side.