To determine the length of one leg of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle with a hypotenuse of 4 cm, follow these steps:
1. Recognize the Properties of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] Triangle:
A [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle (also known as an isosceles right triangle) has two legs of equal length. The relationship between the legs and the hypotenuse in this type of triangle is that each leg is [tex]\(\frac{1}{\sqrt{2}}\)[/tex] times the hypotenuse.
2. Formula Application:
For a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, if the hypotenuse [tex]\(c\)[/tex] is known, each leg [tex]\(a\)[/tex] can be calculated using the formula:
[tex]\[
a = \frac{c}{\sqrt{2}}
\][/tex]
3. Substitute the Given Hypotenuse:
Given that the hypotenuse [tex]\(c = 4\)[/tex] cm, substitute it into the formula:
[tex]\[
a = \frac{4}{\sqrt{2}}
\][/tex]
4. Simplify the Expression:
Simplify [tex]\(\frac{4}{\sqrt{2}}\)[/tex] by multiplying the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex] to rationalize the denominator:
[tex]\[
a = \frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2} = 2\sqrt{2}
\][/tex]
5. Conclusion:
Therefore, the length of one leg of the triangle is [tex]\( 2\sqrt{2} \)[/tex] cm.
Thus, the correct answer is [tex]\(2 \sqrt{2}\)[/tex] cm.
