To determine the length of the hypotenuse in a [tex]\(45^\circ\)[/tex]-[tex]\(45^\circ\)[/tex]-[tex]\(90^\circ\)[/tex] triangle, we can use the properties of this special right triangle. In a [tex]\(45^\circ\)[/tex]-[tex]\(45^\circ\)[/tex]-[tex]\(90^\circ\)[/tex] triangle, the lengths of the legs are equal, and the hypotenuse is related to the legs by the following ratio: the hypotenuse is the length of a leg times [tex]\(\sqrt{2}\)[/tex].
Here we know that each leg of the triangle measures [tex]\(12 \, \text{cm}\)[/tex].
Let's denote the leg length by [tex]\( a \)[/tex]. Thus, we have:
[tex]\[ a = 12 \, \text{cm} \][/tex]
Using the special ratio for a [tex]\(45^\circ\)[/tex]-[tex]\(45^\circ\)[/tex]-[tex]\(90^\circ\)[/tex] triangle, the hypotenuse [tex]\( h \)[/tex] can be calculated as:
[tex]\[ h = a \times \sqrt{2} \][/tex]
[tex]\[ h = 12 \times \sqrt{2} \][/tex]
Given this expression, let's substitute [tex]\(\sqrt{2}\)[/tex] and calculate the value:
[tex]\[ h = 12 \times 1.4142135623730951 \][/tex]
[tex]\[ h \approx 16.970562748477143 \][/tex]
Therefore, the length of the hypotenuse is approximately [tex]\( 16.97 \, \text{cm} \)[/tex].
Among the given options, the correct one that matches this value is:
[tex]\[ 12\sqrt{2} \, \text{cm} \][/tex]
So, the length of the hypotenuse is [tex]\( 12\sqrt{2} \, \text{cm} \)[/tex].
