Certainly! Let's solve the problem step-by-step.
A [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle is a special right triangle. In such a triangle, the two legs are equal in length, and the hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the length of each leg.
Given:
- The hypotenuse is 18 cm
We need to find the length of one leg. Let's denote the length of one leg as [tex]\( \text{leg} \)[/tex].
Using the property of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle:
[tex]\[ \text{hypotenuse} = \text{leg} \times \sqrt{2} \][/tex]
Therefore:
[tex]\[ 18 = \text{leg} \times \sqrt{2} \][/tex]
Solving for [tex]\( \text{leg} \)[/tex]:
[tex]\[ \text{leg} = \frac{18}{\sqrt{2}} \][/tex]
To rationalize the denominator:
[tex]\[ \text{leg} = \frac{18}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} \][/tex]
[tex]\[ \text{leg} = \frac{18 \sqrt{2}}{2} \][/tex]
[tex]\[ \text{leg} = 9 \sqrt{2} \][/tex]
Thus, the length of one leg of the triangle is [tex]\( 9 \sqrt{2} \)[/tex] cm.
Therefore, the correct answer is:
[tex]\[ 9 \sqrt{2} \, \text{cm} \][/tex]
