In a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, the sides follow a specific ratio. This kind of triangle is an isosceles right triangle, meaning both legs have the same length, and the hypotenuse can be calculated using this ratio: If each leg is of length [tex]\(a\)[/tex], then the hypotenuse [tex]\(h\)[/tex] is given by [tex]\(a \sqrt{2}\)[/tex].
Given that each leg of the triangle is 6 cm, we can apply this ratio to find the length of the hypotenuse.
1. Start with the given length of the leg: 6 cm.
2. For a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, the hypotenuse is [tex]\( \text{leg} \times \sqrt{2} \)[/tex].
So, the length of the hypotenuse is:
[tex]\[ 6 \times \sqrt{2} \][/tex]
Numerically evaluating [tex]\( 6 \times \sqrt{2} \)[/tex] yields:
[tex]\[ 6 \times \sqrt{2} = 8.485281374238571 \, \text{cm} \][/tex]
Therefore, the correct answer in the context of the multiple-choice options given is:
[tex]\[ 6 \sqrt{2} \, \text{cm} \][/tex]
