To find the length of one leg of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle with a hypotenuse of 4 cm, we use the properties of this special type of triangle.
In a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, the two legs are of equal length, and each leg is related to the hypotenuse by a specific ratio. The length of each leg can be found using the following relationship:
[tex]\[ \text{Leg length} = \frac{\text{Hypotenuse}}{\sqrt{2}} \][/tex]
Given the hypotenuse is 4 cm, we can substitute this value into the formula:
[tex]\[ \text{Leg length} = \frac{4 \, \text{cm}}{\sqrt{2}} \][/tex]
To simplify, we rationalize the denominator:
[tex]\[ \frac{4 \, \text{cm}}{\sqrt{2}} = \frac{4 \, \text{cm} \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{4 \sqrt{2} \, \text{cm}}{2} = 2 \sqrt{2} \, \text{cm} \][/tex]
Therefore, the length of one leg of the triangle is [tex]\(2 \sqrt{2} \, \text{cm}\)[/tex].
So among the given options:
1. 2 cm
2. [tex]\(2 \sqrt{2} \, \text{cm}\)[/tex]
3. 4 cm
4. [tex]\(4 \sqrt{2} \, \text{cm}\)[/tex]
The correct answer is:
[tex]\[ 2 \sqrt{2} \, \text{cm} \][/tex]
