The easiest way to disprove a conjecture is to find a counter example to the statement. If you suspect that a statement is not necessarily true, find a counter example.
Which of the following statements about triangles are true?
I. The perimeter of a triangle with integer sides is an integer.
II. The area of a triangle with integer sides is an integer.
III. 2 triangles with the same perimeter are similar.
A) I only
B) II only
C) III only
D) I and II only
E) I, II, and III
Solution: Consider the first statement. The perimeter of a triangle is the sum of its 3 sides. Since each of the sides is an integer, hence the perimeter is an integer.
Consider the second statement. This statement looks suspiciously false, but it can be hard to find a counter example. The favorite or right triangles have integer area. We know that the area is equal to half base times height. The base is an integer, so if we can make the height a non-integer, then it's possible to find a counterexample. For what triangles is it easy to find their height? Let's consider an isosceles triangle! The isosceles triangle has a height of . Clearly, this triangle does not have integer area!
Consider the third statement. The triangles and are clearly not similar. Hence it is not true.
Thus, only the first statement is true. The answer is A.