To determine the length of one leg of a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle with a given hypotenuse, we need to use the properties of this special triangle.
In a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle, the legs are of equal length. If we denote the leg length by [tex]\( x \)[/tex], then the relationship between the legs and the hypotenuse [tex]\( h \)[/tex] is given by the formula:
[tex]\[ h = x \sqrt{2} \][/tex]
Given that the hypotenuse [tex]\( h \)[/tex] is 18 cm, we can set up the following equation:
[tex]\[ 18 = x \sqrt{2} \][/tex]
To solve for [tex]\( x \)[/tex], we isolate [tex]\( x \)[/tex] by dividing both sides of the equation by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ x = \frac{18}{\sqrt{2}} \][/tex]
Next, to rationalize the denominator, we multiply the numerator and the denominator by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ x = \frac{18 \sqrt{2}}{2} \][/tex]
This simplifies to:
[tex]\[ x = 9 \sqrt{2} \][/tex]
Thus, the length of one leg of the [tex]\( 45^{\circ}-45^{\circ}-90^{\circ} \)[/tex] triangle is [tex]\( 9 \sqrt{2} \)[/tex] cm. Therefore, the correct answer is:
[tex]\[ 9 \sqrt{2} \text{ cm} \][/tex]
