To solve this problem, let's follow the properties of a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle.
1. Identify the Triangle Properties: A [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle is an isosceles right triangle. In such a triangle, both legs are of equal length, and the hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the length of one leg.
2. Given Information: The hypotenuse measures 4 cm.
3. Apply the Properties: Let [tex]\( L \)[/tex] represent the length of one leg of the triangle. According to the property of the [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle:
[tex]\[ \text{Hypotenuse} = L \cdot \sqrt{2} \][/tex]
4. Set Up the Equation: Since the hypotenuse is given as 4 cm:
[tex]\[ 4 = L \cdot \sqrt{2} \][/tex]
5. 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]
6. Simplify the Expression:
[tex]\[ L = 4 \div \sqrt{2} \][/tex]
7. To rationalize the denominator, multiply the numerator and the denominator by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ L = \frac{4 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{4 \sqrt{2}}{2} = 2 \sqrt{2} \][/tex]
8. Conclusion: The length of one leg of the triangle is:
[tex]\[ 2 \sqrt{2} \, \text{cm} \][/tex]
Thus, the answer is [tex]\( 2 \sqrt{2} \, \text{cm} \)[/tex].
