Is the perimeter of triangle ABC less than 15 centimetres ?

Official Solution:


Is the perimeter of triangle ABC less than 15 centimetres ?

To solve this question recall the following property: the length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.

(1) The lengths of the sides of triangle ABC are consecutive even integers in centimetres

The smallest three consecutive positive even integers are 2, 4, and 6 but a triangle cannot have sides equal to 2, 4, and 6 centimetres because in this case the length of one side (6) is NOT smaller than the sum of two other sides (2 and 4). The next three consecutive positive even integers are 4, 6, and 8. The sum is 18. So, the perimeter of triangle ABC must be greater than 15 centimetres . Sufficient.

(2) The sum of the lengths of the shortest and longest sides of triangle ABC is 12 centimetres

The perimeter to be less than 15 centimetres, the third side (second smallest) must be less than 3 centimetres. In this case the sides would be: (less than 3), (less than 3), (more than 9). But since the sum of two sides is NOT greater than the third side ((less than 3) + (less than 3) < 9), than such a triangle is not possible. So, the perimeter of triangle ABC must be greater than 15 centimetres . Sufficient.


Answer: D