To solve for the length of one leg of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle where the hypotenuse is given as 4 cm, we can use properties specific to this type of right triangle.
1. Understand the Triangle: In a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, the two legs are of equal length since the angles opposite these legs are both [tex]\(45^\circ\)[/tex].
2. Relationship Between the Legs and Hypotenuse: The hypotenuse ([tex]\(c\)[/tex]) of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle is equal to the leg length ([tex]\(a\)[/tex]) multiplied by [tex]\(\sqrt{2}\)[/tex]. This can be expressed as:
[tex]\[
c = a\sqrt{2}
\][/tex]
3. Plug in the Given Hypotenuse: Here, the hypotenuse ([tex]\(c\)[/tex]) is 4 cm. Thus, we set up the equation:
[tex]\[
4 = a\sqrt{2}
\][/tex]
4. Solve for the Leg Length [tex]\(a\)[/tex]: To find the length of one of the legs, solve for [tex]\(a\)[/tex]:
[tex]\[
a = \frac{4}{\sqrt{2}}
\][/tex]
5. Rationalize the Denominator (optional but often done in math): To rationalize the denominator, multiply by [tex]\(\frac{\sqrt{2}}{\sqrt{2}}\)[/tex]:
[tex]\[
a = \frac{4 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{4\sqrt{2}}{2} = 2\sqrt{2}
\][/tex]
Thus, the length of one leg of the triangle is:
[tex]\[
2\sqrt{2} \text{ cm}
\][/tex]
Therefore, the correct answer is:
[tex]\[
2 \sqrt{2} \text{ cm}
\][/tex]
