Which of the following lengths can be the sidelengths of a triangle? Explain your answers. Note that for each set, the three lengths have the same unit of measure. (a) 2, 2, 5 (b) 3, 4, 6 (c) 2, 4, 8 (d) 5, 5, 8 (e) 10, 20, 25 (f) 10, 20, 35 (g) 24, 26, 28
No triangle exists. Longest side 5 > 2 + 2 = 4.
Triangle exists. Longest side 6 < 3 + 4 = 7.
No triangle exists. Longest side 8 > 2 + 4 = 6.
Triangle exists. Longest side 8 < 5 + 5 = 10.
Triangle exists. Longest side 25 < 10 + 20 = 30.
No triangle exists. Longest side 35 > 10 + 20 = 30.
Triangle exists. Longest side 28 < 24 + 26 = 50.
Explanation
(a) Triangle inequality violated. (b) Triangle inequality satisfied. (c) Triangle inequality violated. (d) Triangle inequality satisfied. (e) Triangle inequality satisfied. (f) Triangle inequality violated. (g) Triangle inequality satisfied.