Which set of line segments can be used to construct a triangle?
Responses
A 8 cm, 3 cm, and 3 cm
B 6 cm, 5 cm, and 2 cm
C 6 cm, 2 cm, and 9 cm
D 8 cm, 2 cm, and 5 cm
B 6 cm, 5 cm, and 2 cm
why
To satisfy the triangle inequality, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In option B, 6 + 5 > 2 and 6 + 2 > 5 and 5 + 2 > 6, so these three line segments can be used to construct a triangle.
In option A, 8 + 3 < 3, which violates the triangle inequality.
In option C, 6 + 2 < 9, which also violates the triangle inequality.
In option D, 8 + 2 < 5, which again violates the triangle inequality.
To determine which set of line segments can be used to construct a triangle, we need to apply the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
Let's check each set of line segments:
A) 8 cm, 3 cm, and 3 cm: The sum of the two shorter sides (3 cm and 3 cm) is 6 cm, which is equal to the length of the longest side (8 cm). Since the sum of the two shorter sides is not greater than the length of the longest side, this set of line segments cannot be used to construct a triangle.
B) 6 cm, 5 cm, and 2 cm: The sum of the two shorter sides (2 cm and 5 cm) is 7 cm, which is greater than the length of the longest side (6 cm). Therefore, this set of line segments can be used to construct a triangle.
C) 6 cm, 2 cm, and 9 cm: The sum of the two shorter sides (2 cm and 6 cm) is 8 cm, which is greater than the length of the longest side (9 cm). Thus, this set of line segments can be used to construct a triangle.
D) 8 cm, 2 cm, and 5 cm: The sum of the two shorter sides (2 cm and 5 cm) is 7 cm, which is greater than the length of the longest side (8 cm). So, this set of line segments can also be used to construct a triangle.
Therefore, the correct answers are B) 6 cm, 5 cm, and 2 cm, and C) 6 cm, 2 cm, and 9 cm.