Question

Which of the following side measurements could make an equilateral triangle and isosceles triangle?

4, 5, 6
2, 2, 4
2, 2, 2
12, 12, 14

Answer 1

Check each triple using the triangle inequality (sum of any two sides must be greater than the third) and the side-equality tests.

- 4, 5, 6: 4+5>6, 4+6>5, 5+6>4 — forms a triangle, but all sides different → scalene (not isosceles or equilateral).
- 2, 2, 4: 2+2 = 4 (not > 4) → degenerate, does not form a (nondegenerate) triangle.
- 2, 2, 2: all sides equal → equilateral (and therefore also isosceles).
- 12, 12, 14: 12+12>14, etc. → forms a triangle with two equal sides → isosceles.

So only 2,2,2 can make an equilateral triangle; 2,2,2 and 12,12,14 can make isosceles triangles.