To determine the length of one leg of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle given that the hypotenuse measures 128 cm, we can follow these steps:
1. Recognize the properties of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle. In this type of triangle, the lengths of the legs are equal. Furthermore, the relationship between the leg length (we'll call this [tex]\(a\)[/tex]) and the hypotenuse ([tex]\(c\)[/tex]) can be derived from the Pythagorean theorem. The formula for the hypotenuse [tex]\(c\)[/tex] in terms of the leg length [tex]\(a\)[/tex] is given by:
[tex]\[
c = a\sqrt{2}
\][/tex]
Given:
[tex]\[
c = 128 \text{ cm}
\][/tex]
2. Rearrange the formula to solve for the leg length [tex]\(a\)[/tex]:
[tex]\[
a = \frac{c}{\sqrt{2}}
\][/tex]
3. Substitute the known value of the hypotenuse [tex]\(c = 128\)[/tex] cm into the equation:
[tex]\[
a = \frac{128}{\sqrt{2}}
\][/tex]
4. Rationalize the denominator:
[tex]\[
a = \frac{128 \sqrt{2}}{2}
\][/tex]
5. Simplify the expression:
[tex]\[
a = 64 \sqrt{2} \text{ cm}
\][/tex]
Thus, the length of one leg of the [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle is:
[tex]\[
64 \sqrt{2} \text{ cm}
\][/tex]
Therefore, the correct choice from the given options is:
[tex]\[
\boxed{64 \sqrt{2} \text{ cm}}
\][/tex]
