Question 447811
That statement is not true; it is possible for the perimeter of an isosceles triangle to be a whole number.


Suppose the sides are *[tex x], *[tex x], and *[tex x \sqrt{2}], where x is any positive number. The perimeter is therefore


*[tex \LARGE P = 2x + x \sqrt{2} = x(2 + \sqrt{2})].


Suppose *[tex \LARGE x = \frac{5}{2 + \sqrt{2}}]. If we assume such a value of x, then the perimeter would be


*[tex \LARGE P = \frac{5}{2 + \sqrt{2}} (2 + \sqrt{2}) = 5], which is an integer. In fact, we can replace the 5 in the numerator of x with any integer to obtain an integer perimeter.


However, if the side length itself was an integer, then it would be impossible. Proving this simply relies on the fact that *[tex \LARGE \sqrt{2}] is irrational and cannot be multiplied by any rational number to obtain a rational number.