In a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, the two legs are equal in length, and the relationship between the legs and the hypotenuse follows the ratio [tex]\(1:1:\sqrt{2}\)[/tex]. This means that if the length of each leg is [tex]\(14\)[/tex] cm, the hypotenuse ([tex]\(c\)[/tex]) can be found by multiplying the leg length by [tex]\(\sqrt{2}\)[/tex].
Here's the step-by-step process to determine the length of the hypotenuse:
1. Identify the length of the legs: Each leg of the triangle is [tex]\(14\)[/tex] cm.
2. Understand the relationship in a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle: The hypotenuse is equal to the leg length times [tex]\(\sqrt{2}\)[/tex].
[tex]\[
\text{Hypotenuse} = \text{leg length} \times \sqrt{2}
\][/tex]
3. Substitute the given leg length into the formula:
[tex]\[
\text{Hypotenuse} = 14 \, \text{cm} \times \sqrt{2}
\][/tex]
4. Calculate the numerical value:
[tex]\[
14 \times \sqrt{2} \approx 19.79898987322333 \, \text{cm}
\][/tex]
Therefore, the length of the hypotenuse is approximately [tex]\(19.8\)[/tex] cm, rounded to one decimal place. So, the correct choice from the given options is:
[tex]\[
14 \sqrt{2} \, \text{cm}
\][/tex]
