Question

question 6 of 10
a triangle has two sides of lengths 6 and 9. what value could the length of the third side be? check all that apply.
a. 2
b. 12
c. 4
d. 15
e. 10
f. 7

Explanation:

Step1: Use the triangle inequality theorem

For a triangle with side lengths \(a\), \(b\), and \(c\), the following must hold: \(|a - b|\lt c\lt a + b\). Here \(a = 6\) and \(b=9\).
First, calculate \(|a - b|\): \(|6 - 9|=3\). Then calculate \(a + b\): \(6+9 = 15\). So \(3\lt c\lt15\).

Step2: Check each option

• Option A: \(2\lt3\), does not satisfy \(3\lt c\lt15\).
• Option B: \(3\lt12\lt15\), satisfies \(3\lt c\lt15\).
• Option C: \(3\lt4\lt15\), satisfies \(3\lt c\lt15\).
• Option D: \(15 = 15\), does not satisfy \(3\lt c\lt15\).
• Option E: \(3\lt10\lt15\), satisfies \(3\lt c\lt15\).
• Option F: \(3\lt7\lt15\), satisfies \(3\lt c\lt15\).

Answer:

B. 12, C. 4, E. 10, F. 7