Q1.
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
Answer
Compare the longest length with the sum of the other two in each set.
| Set | Longest vs sum of other two | Sidelengths of a triangle? |
|---|---|---|
| (a) 2, 2, 5 | 5 > 2 + 2 = 4 | No |
| (b) 3, 4, 6 | 6 < 3 + 4 = 7 | Yes |
| (c) 2, 4, 8 | 8 > 2 + 4 = 6 | No |
| (d) 5, 5, 8 | 8 < 5 + 5 = 10 | Yes |
| (e) 10, 20, 25 | 25 < 10 + 20 = 30 | Yes |
| (f) 10, 20, 35 | 35 > 10 + 20 = 30 | No |
| (g) 24, 26, 28 | 28 < 24 + 26 = 50 | Yes |
Writing out all three comparisons for the successful sets:
(b) 3 < 4 + 6, 4 < 3 + 6, 6 < 3 + 4 ✓
(d) 5 < 5 + 8, 5 < 5 + 8, 8 < 5 + 5 ✓
(e) 10 < 20 + 25, 20 < 10 + 25, 25 < 10 + 20 ✓
(g) 24 < 26 + 28, 26 < 24 + 28, 28 < 24 + 26 ✓
(d) 5 < 5 + 8, 5 < 5 + 8, 8 < 5 + 5 ✓
(e) 10 < 20 + 25, 20 < 10 + 25, 25 < 10 + 20 ✓
(g) 24 < 26 + 28, 26 < 24 + 28, 28 < 24 + 26 ✓
Why it happens: Sets (a), (c) and (f) each have one length that is larger than the other two put together, so those two sides can never reach across it. Sets (b), (d), (e) and (g) satisfy the triangle inequality, so the two arcs are certain to cross and the triangle certainly exists.
Tip: (d) 5, 5, 8 is isosceles and (g) 24, 26, 28 is scalene — the inequality decides existence, not the type.