Apply the Triangle Inequality Theorem to determine which three side lengths form a triangle.
A. 2, 3, 6
B. 7, 5, 13
C. 7, 8, 9
D. 4, 8, 12
Using the Triangle Inequality Theorem, a triangle can be formed if and only if the sum of any two side lengths is greater than the third side length.
A. For the side lengths 2, 3, and 6:
2 + 3 = 5, which is less than 6.
3 + 6 = 9, which is greater than 2.
2 + 6 = 8, which is greater than 3.
Therefore, a triangle cannot be formed with side lengths 2, 3, and 6.
B. For the side lengths 7, 5, and 13:
7 + 5 = 12, which is less than 13.
5 + 13 = 18, which is greater than 7.
7 + 13 = 20, which is greater than 5.
Therefore, a triangle can be formed with side lengths 7, 5, and 13.
C. For the side lengths 7, 8, and 9:
7 + 8 = 15, which is greater than 9.
8 + 9 = 17, which is greater than 7.
7 + 9 = 16, which is greater than 8.
Therefore, a triangle can be formed with side lengths 7, 8, and 9.
D. For the side lengths 4, 8, and 12:
4 + 8 = 12, which is equal to 12.
8 + 12 = 20, which is greater than 4.
4 + 12 = 16, which is greater than 8.
Therefore, a triangle can be formed with side lengths 4, 8, and 12.
In conclusion, the three side lengths that form a triangle are:
- 7, 5, 13 (option B)
- 7, 8, 9 (option C)
- 4, 8, 12 (option D)