To determine the length of one leg of a [tex]\(45^\circ\)[/tex]-[tex]\(45^\circ\)[/tex]-[tex]\(90^\circ\)[/tex] triangle given that the hypotenuse measures 4 cm, we need to use the properties of this special type of triangle.
A [tex]\(45^\circ\)[/tex]-[tex]\(45^\circ\)[/tex]-[tex]\(90^\circ\)[/tex] triangle is an isosceles right triangle, meaning the two legs are congruent. Additionally, the relationship between the legs and the hypotenuse in this triangle is given by:
[tex]\[ \text{Leg length} = \frac{\text{Hypotenuse}}{\sqrt{2}} \][/tex]
Given:
- Hypotenuse = 4 cm
Using the relationship:
[tex]\[ \text{Leg length} = \frac{4}{\sqrt{2}} \][/tex]
To simplify this expression, multiply the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ \text{Leg length} = \frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2} = 2\sqrt{2} \][/tex]
So, the length of one leg of the triangle is [tex]\(2\sqrt{2}\)[/tex] cm.
Therefore, the correct answer is:
[tex]\[ \boxed{2 \sqrt{2} \text{ cm}} \][/tex]
