Sure, I'd be happy to help with this problem!
To start, note that a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle is an isosceles right triangle where both legs are of equal length. In such a triangle, the relationship between the hypotenuse [tex]\(h\)[/tex] and each leg [tex]\(l\)[/tex] is given by the following formula:
[tex]\[ h = l \sqrt{2} \][/tex]
Given that the hypotenuse [tex]\(h\)[/tex] measures [tex]\(128 \, \text{cm}\)[/tex], we need to find the length of one leg [tex]\(l\)[/tex]. We can isolate [tex]\(l\)[/tex] by rearranging the equation:
[tex]\[ l = \frac{h}{\sqrt{2}} \][/tex]
Substitute the given hypotenuse value into the equation:
[tex]\[ l = \frac{128 \, \text{cm}}{\sqrt{2}} \][/tex]
Solving this expression, we find the length of one leg:
[tex]\[ l \approx 90.50966799187808 \, \text{cm} \][/tex]
Therefore, the correct answer from the provided choices is:
None of the given options is exactly [tex]\(90.50966799187808 \, \text{cm}\)[/tex], however, this is the correct length of one leg.
So, the length of one leg of the triangle is approximately [tex]\(90.51 \, \text{cm}\)[/tex].
