To find the length of one leg of a [tex]\( 45^{\circ}-45^{\circ}-90^{\circ} \)[/tex] triangle given that the hypotenuse is 4 cm, we can use the properties of this special right triangle.
In a [tex]\( 45^{\circ}-45^{\circ}-90^{\circ} \)[/tex] triangle (also known as an isosceles right triangle):
- The legs are of equal length.
- The relationship between the legs and the hypotenuse is given by the formula:
[tex]\[
\text{leg} = \frac{\text{hypotenuse}}{\sqrt{2}}
\][/tex]
Given that the hypotenuse is 4 cm, we can now substitute this value into the formula:
[tex]\[
\text{leg} = \frac{4 \text{ cm}}{\sqrt{2}}
\][/tex]
To rationalize the denominator, multiply both numerator and denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
\text{leg} = \frac{4 \sqrt{2} \text{ cm}}{2}
\][/tex]
Simplifying this expression:
[tex]\[
\text{leg} = 2 \sqrt{2} \text{ cm}
\][/tex]
Therefore, the length of one leg of the triangle is:
[tex]\[
2 \sqrt{2} \text{ cm}
\][/tex]
The correct answer is:
[tex]\[
2 \sqrt{2} \text{ cm}
\][/tex]
