To find the length of one leg of a \(45^\circ-45^\circ-90^\circ\) triangle with a hypotenuse of 18 cm, we can leverage the properties of this special right triangle.
In a \(45^\circ-45^\circ-90^\circ\) triangle, the relationship between the hypotenuse and the legs is well defined. Specifically:
- The legs of the triangle are congruent (they have the same length).
- Each leg is equal to the hypotenuse divided by \(\sqrt{2}\).
Given the hypotenuse \(c = 18\) cm, we need to calculate the length of one leg. The formula for the leg length \(a\) in a \(45^\circ-45^\circ-90^\circ\) triangle is:
[tex]\[ a = \frac{c}{\sqrt{2}} \][/tex]
Substituting the given hypotenuse value into the formula, we get:
[tex]\[ a = \frac{18}{\sqrt{2}} \][/tex]
Simplifying the expression, we find:
[tex]\[ a = \frac{18}{\sqrt{2}} \approx 12.727922061357855 \, \text{cm} \][/tex]
Thus, the length of one leg of the triangle is approximately \(12.73 \, \text{cm}\). However, none of the answer choices given in the problem list this option directly. So, taking a closer look, the most fitting answer according to the values provided would be approximately \(12.73 \, \text{cm}\).
Therefore, the correct choice is not listed explicitly in the provided options, but considering common misinterpretations:
- \(9 \, \text{cm}\) is incorrect.
- \(9 \sqrt{2} \, \text{cm}\) is also incorrect since \(9 \sqrt{2} \approx 12.73\).
Thus, if we had to strictly choose from the provided options, none would be exact, but interpretively close values might be [tex]\(\boxed{9 \sqrt{2} \, \text{cm}}\)[/tex]. However, precision is key, and the correct calculated value for one leg is approximately [tex]\(12.73 \, \text{cm}\)[/tex].
