To determine the length of one of the legs of a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle when the hypotenuse is given, we can use the properties of this special type of triangle.
1. In a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle, the legs are of equal length.
2. The relationship between the legs and the hypotenuse in this type of triangle is given by the ratio [tex]\( 1:\sqrt{2} \)[/tex]. This means that if the length of each leg is [tex]\( x \)[/tex], then the hypotenuse is [tex]\( x\sqrt{2} \)[/tex].
Given that the hypotenuse measures 24 inches, we need to find the length of one leg. Let's denote the length of one leg as [tex]\( x \)[/tex].
Considering the properties of the [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle:
[tex]\[ x\sqrt{2} = 24 \][/tex]
To find [tex]\( x \)[/tex], divide both sides of the equation by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ x = \frac{24}{\sqrt{2}} \][/tex]
To rationalize the denominator, multiply the numerator and the denominator by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ x = \frac{24 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} \][/tex]
[tex]\[ x = \frac{24 \cdot \sqrt{2}}{2} \][/tex]
[tex]\[ x = 12\sqrt{2} \][/tex]
So, the length of one of the legs of the [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle is [tex]\(\boxed{12\sqrt{2} \, \text{inches}}\)[/tex].
