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Counter Examples

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 \( 3-4-5\) or \( 5-12-13\) 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 \( 1-2-2\) isosceles triangle has a height of \( \sqrt{2^2 - .5^2 } = \frac{1}{2} \sqrt{17} \). Clearly, this triangle does not have integer area!

Consider the third statement. The triangles \( 3-4-5\) and \(4-4-4\) are clearly not similar. Hence it is not true.

Thus, only the first statement is true. The answer is A.

Note by Arron Kau
2 years, 2 months ago

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