To find the length of one leg of a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle when the hypotenuse is given, we can use specific properties of this type of triangle.
A [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle is also known as an isosceles right triangle. In such triangles, the legs are congruent and each leg is related to the hypotenuse by the following relationship:
[tex]\[ \text{leg} = \frac{\text{hypotenuse}}{\sqrt{2}} \][/tex]
Given:
[tex]\[ \text{hypotenuse} = 4 \text{ cm} \][/tex]
Using the relationship:
[tex]\[ \text{leg} = \frac{4 \text{ cm}}{\sqrt{2}} \][/tex]
To simplify [tex]\(\frac{4}{\sqrt{2}}\)[/tex], we can multiply the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex] for rationalizing denominators:
[tex]\[ \frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2} = 2\sqrt{2} \text{ cm} \][/tex]
So, the length of one leg of the triangle is:
[tex]\[ 2\sqrt{2} \text{ cm} \][/tex]
Hence, the correct answer is:
[tex]\[ 2\sqrt{2} \text{ cm} \][/tex]
