Answer :
To determine whether a triangle can be formed using side lengths 6, 9, and 12, we must apply the triangle inequality theorem. This theorem states that for three side lengths to form a triangle, the sum of any two sides must be greater than the third side.
Let's check each condition:
1. First condition: The sum of the first side and the second side must be greater than the third side:
[tex]\[ 6 + 9 > 12 \][/tex]
Simplifying,
[tex]\[ 15 > 12 \][/tex]
This condition is true.
2. Second condition: The sum of the first side and the third side must be greater than the second side:
[tex]\[ 6 + 12 > 9 \][/tex]
Simplifying,
[tex]\[ 18 > 9 \][/tex]
This condition is true.
3. Third condition: The sum of the second side and the third side must be greater than the first side:
[tex]\[ 9 + 12 > 6 \][/tex]
Simplifying,
[tex]\[ 21 > 6 \][/tex]
This condition is true.
Since all three conditions of the triangle inequality theorem are satisfied, we can conclude that a triangle can be formed with side lengths 6, 9, and 12.
Therefore, the correct answer is:
Yes, because [tex]\(6 + 9 > 12\)[/tex].
Let's check each condition:
1. First condition: The sum of the first side and the second side must be greater than the third side:
[tex]\[ 6 + 9 > 12 \][/tex]
Simplifying,
[tex]\[ 15 > 12 \][/tex]
This condition is true.
2. Second condition: The sum of the first side and the third side must be greater than the second side:
[tex]\[ 6 + 12 > 9 \][/tex]
Simplifying,
[tex]\[ 18 > 9 \][/tex]
This condition is true.
3. Third condition: The sum of the second side and the third side must be greater than the first side:
[tex]\[ 9 + 12 > 6 \][/tex]
Simplifying,
[tex]\[ 21 > 6 \][/tex]
This condition is true.
Since all three conditions of the triangle inequality theorem are satisfied, we can conclude that a triangle can be formed with side lengths 6, 9, and 12.
Therefore, the correct answer is:
Yes, because [tex]\(6 + 9 > 12\)[/tex].
