Question

the lengths of the sides of a triangle can be 3 feet, 5 feet, and feet.

options:
3
16
15
8

Explanation:

🆕 New Concept Discovered: Triangle Inequality Theorem
Any two sides must add up to more than the third side.

Step 1: Understand the rule

For any triangle with side lengths \(a\), \(b\), and \(c\), the sum of the lengths of any two sides must be strictly greater than the length of the remaining side:

$$a + b > c$$

$$a + c > b$$

$$b + c > a$$

This means the third side \(x\) must be strictly between the difference and the sum of the two known sides:

$$|a - b| < x < a + b$$

Step 2: Calculate the possible range

Given the two known sides are \(3\) feet and \(5\) feet:

• Minimum limit: \(5 - 3 = 2\) feet
• Maximum limit: \(5 + 3 = 8\) feet

So, the third side \(x\) must satisfy:

$$2 < x < 8$$

Step 3: Test the given options

We check which of the options (\(3\), \(16\), \(15\), \(8\)) falls strictly between \(2\) and \(8\):

• \(3\): Since \(2 < 3 < 8\), this is a valid side length.
• \(16\): Too large (\(16 \ge 8\)).
• \(15\): Too large (\(15 \ge 8\)).
• \(8\): Too large (\(8\) is not strictly less than \(8\)).

Answer:

3