To solve for the length of one leg in a 45°-45°-90° triangle where the hypotenuse measures 4 cm, we can utilize the special properties of this type of triangle.
A 45°-45°-90° triangle, also known as an isosceles right triangle, has sides that follow a specific ratio:
- The legs are of equal length.
- The hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the length of each leg.
Given the hypotenuse ([tex]\(c\)[/tex]) is 4 cm, we can denote the length of each leg as [tex]\(a\)[/tex]. According to the relationship in a 45°-45°-90° triangle, the hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the leg:
[tex]\[ c = a \sqrt{2} \][/tex]
Substitute the given hypotenuse length into the equation:
[tex]\[ 4 = a \sqrt{2} \][/tex]
To find the length of the leg [tex]\(a\)[/tex], solve for [tex]\(a\)[/tex]:
[tex]\[ a = \frac{4}{\sqrt{2}} \][/tex]
Rationalize the denominator by multiplying the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ a = \frac{4 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} \][/tex]
[tex]\[ a = \frac{4 \sqrt{2}}{2} \][/tex]
[tex]\[ a = 2 \sqrt{2} \][/tex]
Thus, the length of one leg of the triangle is:
[tex]\[ 2 \sqrt{2} \text{ cm} \][/tex]
So the correct answer is:
[tex]\[
\boxed{2 \sqrt{2} \text{ cm}}
\][/tex]
