To solve this problem, we need to understand the properties of a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle. In such a triangle, the legs are of equal length and are both denoted as [tex]\(a\)[/tex]. The relationship between the legs and the hypotenuse (denoted as [tex]\(c\)[/tex]) in a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle is given by:
[tex]\[ c = a \sqrt{2} \][/tex]
Given that the hypotenuse [tex]\( c = 128 \)[/tex] cm, we can find the length of one leg [tex]\( a \)[/tex] by solving the equation:
[tex]\[ 128 = a \sqrt{2} \][/tex]
To isolate [tex]\( a \)[/tex], divide both sides by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ a = \frac{128}{\sqrt{2}} \][/tex]
To simplify this expression, we multiply the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ a = \frac{128 \sqrt{2}}{2} \][/tex]
[tex]\[ a = 64 \sqrt{2} \][/tex]
Therefore, the length of one leg of the triangle is:
[tex]\[\boxed{64 \sqrt{2} \, \text{cm}}\][/tex]
