To determine the length of one leg of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle given the hypotenuse of [tex]\(128 \, \text{cm}\)[/tex], we need to use the properties of this type of triangle.
In a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, the legs are congruent, and the length of each leg can be derived from the hypotenuse by the following relationship:
[tex]\[ \text{leg length} = \frac{\text{hypotenuse}}{\sqrt{2}} \][/tex]
Given:
[tex]\[ \text{hypotenuse} = 128 \, \text{cm} \][/tex]
We can substitute this value into our formula:
[tex]\[ \text{leg length} = \frac{128 \, \text{cm}}{\sqrt{2}} \][/tex]
Rather than involving the numerical details here, we know from the pre-calculated answer that:
[tex]\[ \frac{128}{\sqrt{2}} \approx 90.50966799187808 \, \text{cm} \][/tex]
Therefore, the exact leg length of the triangle is approximately [tex]\(90.50966799187808 \, \text{cm}\)[/tex].
Given the options presented:
- [tex]\(64 \, \text{cm}\)[/tex]
- [tex]\(64 \sqrt{2} \, \text{cm}\)[/tex]
- [tex]\(128 \, \text{cm}\)[/tex]
- [tex]\(128 \sqrt{2} \, \text{cm}\)[/tex]
None of these options except for [tex]\(90.50966799187808 \, \text{cm}\)[/tex] match the calculated leg length. It appears none of the given multiple-choice options directly correspond to the correct answer, which is:
[tex]\[ \boxed{90.50966799187808 \, \text{cm}} \][/tex]
