To determine the length of one leg of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle where the hypotenuse is given as 128 cm, we need to understand the properties of such a triangle.
1. Basics of 45-45-90 Triangle:
- In a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, the legs are of equal length.
- Each leg has a length equal to the hypotenuse divided by the square root of 2.
2. Explanation:
- Let [tex]\( h \)[/tex] be the length of the hypotenuse.
Given:
[tex]\[
h = 128 \text{ cm}
\][/tex]
- Let [tex]\( l \)[/tex] be the length of one leg of the triangle.
According to the properties, we have:
[tex]\[
l = \frac{h}{\sqrt{2}}
\][/tex]
3. Calculation:
- Substitute the given value of [tex]\( h \)[/tex]:
[tex]\[
l = \frac{128}{\sqrt{2}}
\][/tex]
- To simplify [tex]\( l \)[/tex], multiply both the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
l = \frac{128 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{128 \sqrt{2}}{2} = 64 \sqrt{2}
\][/tex]
4. Verification:
Referring to the numerical result obtained:
[tex]\[
l \approx 90.50966799187808 \text{ cm}
\][/tex]
Notice that this corresponds closely to [tex]\( 64 \sqrt{2} \)[/tex] cm.
Thus, the length of one leg of the [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle is:
[tex]\[
\boxed{64 \sqrt{2} \text{ cm}}
\][/tex]
So, the correct choice from the given options is:
[tex]\[
64 \sqrt{2} \text{ cm}
\][/tex]
