Question

how does the figure help verify the triangle inequality theorem?
a. by showing that only one triangle can be formed when the sum of the lengths of two sides equals the length of the third
b. by showing that only one triangle can be formed when the sum of the lengths of two sides is less than the length of the third
c. by showing that a triangle cannot be formed when the sum of the lengths of two sides is less than the length of the third side
d. by showing that a triangle cannot be formed when the sum of the lengths of two sides equals the length of the third

Explanation:

Step1: Recall triangle inequality theorem

The triangle inequality theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Step2: Analyze the given side lengths

Given side lengths \(7\), \(4\), and \(15\). Calculate \(7 + 4=11\). Since \(11<15\) (sum of two sides \(7\) and \(4\) is less than the third side \(15\)).

Step3: Check each option

• Option A: Incorrect. When \(a + b=c\) (sum of two sides equals the third), a degenerate “triangle” (a straight - line) is formed, not a non - degenerate triangle.
• Option B: Incorrect. The statement in the option is wrong. When \(a + b < c\), no triangle is formed.
• Option C: Correct. As \(7+4 = 11<15\), no triangle is formed, which verifies the triangle inequality theorem (if \(a + b < c\), triangle cannot be formed).
• Option D: Incorrect. When \(a + b=c\), a degenerate “triangle” (a straight - line) is formed, not a non - degenerate triangle.

Answer:

C. by showing that a triangle cannot be formed when the sum of the lengths of two sides is less than the length of the third side