To find 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. Understand the properties of a [tex]\(45^\circ - 45^\circ - 90^\circ\)[/tex] triangle:
- In this type of triangle, the two legs are of equal length.
- The relationship between the legs ([tex]\(L\)[/tex]) and the hypotenuse ([tex]\(H\)[/tex]) is given by the formula: [tex]\( H = L \sqrt{2} \)[/tex].
2. Given information:
- Hypotenuse ([tex]\(H\)[/tex]) = 4 cm.
3. Set up the relationship:
- Since [tex]\( H = L \sqrt{2} \)[/tex], substitution gives us:
[tex]\[ 4 = L \sqrt{2} \][/tex]
4. Solve for [tex]\(L\)[/tex]:
- Isolate [tex]\(L\)[/tex] by dividing both sides of the equation by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ L = \frac{4}{\sqrt{2}} \][/tex]
5. Simplify [tex]\(\frac{4}{\sqrt{2}}\)[/tex]:
- To simplify the expression, multiply the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
L = \frac{4}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2}
\][/tex]
- Simplify the fraction:
[tex]\[
L = \frac{4\sqrt{2}}{2} = 2\sqrt{2}
\][/tex]
6. Conclude the result:
- The length of one leg of the triangle is [tex]\( 2\sqrt{2} \)[/tex] cm.
Hence, the correct answer is:
[tex]\[ \boxed{2\sqrt{2} \, \text{cm}} \][/tex]
