To determine the length of one leg of a [tex]\(45°-45°-90°\)[/tex] triangle with a given hypotenuse of 128 cm, we follow these steps:
1. Understand the Properties of a [tex]\(45°-45°-90°\)[/tex] Triangle:
A [tex]\(45°-45°-90°\)[/tex] triangle is an isosceles right triangle. In such triangles, the legs are congruent, and the relationship between the hypotenuse (let's call it [tex]\(c\)[/tex]) and each leg (let's call them [tex]\(a\)[/tex]) is given by:
[tex]\[
c = a\sqrt{2}
\][/tex]
2. Relate the Hypotenuse to the Leg:
Given that the hypotenuse [tex]\(c\)[/tex] is 128 cm, we can rewrite this relationship to solve for [tex]\(a\)[/tex]:
[tex]\[
a = \frac{c}{\sqrt{2}}
\][/tex]
3. Substitute the Known Value:
Substituting [tex]\(c = 128\)[/tex] cm into the equation:
[tex]\[
a = \frac{128}{\sqrt{2}}
\][/tex]
4. Calculate the Length of one Leg:
To find [tex]\(a\)[/tex], we need the numerical value of [tex]\(\frac{128}{\sqrt{2}}\)[/tex]. This value calculates to approximately 90.50966799187808 cm.
Given the provided options:
- 64 cm
- [tex]\(64\sqrt{2}\)[/tex] cm
- 128 cm
- [tex]\(128\sqrt{2}\)[/tex] cm
We can see that none of the provided options directly match the exact numerical value of the leg length, which is approximately 90.50966799187808 cm. However, to match the context of multiple-choice formatted questions usually presented for such problems, it's important to recognize how those options relate to the geometry of the triangle:
- [tex]\(64\sqrt{2}\)[/tex] cm evaluates to approximately 90.50966799187809 cm, which is close to our result.
Therefore, the length of one leg of the triangle, based on the given hypotenuse of 128 cm, is best matched by the option:
[tex]\[ \boxed{64\sqrt{2} \text{ cm}} \][/tex]
