To determine whether it is possible to construct a triangle with side lengths of 3, 3, and 9, we need to apply the triangle inequality theorem. This theorem states that for three lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. We will check the following three conditions:
1. The sum of the first and second sides must be greater than the third side:
[tex]\(3 + 3 > 9\)[/tex]
2. The sum of the second and third sides must be greater than the first side:
[tex]\(3 + 9 > 3\)[/tex]
3. The sum of the first and third sides must be greater than the second side:
[tex]\(3 + 9 > 3\)[/tex]
Let's evaluate these conditions one by one:
1. [tex]\(3 + 3 = 6\)[/tex], and [tex]\(6\)[/tex] is not greater than [tex]\(9\)[/tex]. So this condition is not satisfied.
2. [tex]\(3 + 9 = 12\)[/tex], and [tex]\(12\)[/tex] is greater than [tex]\(3\)[/tex]. This condition is satisfied.
3. [tex]\(3 + 9 = 12\)[/tex], and [tex]\(12\)[/tex] is greater than [tex]\(3\)[/tex]. This condition is satisfied.
Since the first condition is not met, the side lengths 3, 3, and 9 do not satisfy the triangle inequality theorem. Therefore, it is not possible to form a triangle with these side lengths.
So, the correct answer is:
B. False
