Let's solve the problem step-by-step. You are given a [tex]\(45^\circ - 45^\circ - 90^\circ\)[/tex] triangle, where the hypotenuse measures 12 inches. In a [tex]\(45^\circ - 45^\circ - 90^\circ\)[/tex] triangle, the relationship between the legs and the hypotenuse is well defined:
[tex]\[ \text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}} \][/tex]
Given:
[tex]\[ \text{Hypotenuse} = 12 \text{ inches} \][/tex]
Substitute the hypotenuse value into the formula:
[tex]\[ \text{Leg} = \frac{12}{\sqrt{2}} \][/tex]
Next, simplify this expression. You can multiply the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex] to rationalize the denominator:
[tex]\[ \text{Leg} = \frac{12}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{12 \sqrt{2}}{2} = 6 \sqrt{2} \][/tex]
Hence, the exact length of one leg is:
[tex]\[ 6 \sqrt{2} \text{ inches} \][/tex]
To provide a numerical approximation of this length:
[tex]\[ 6 \sqrt{2} \approx 8.48528137423857 \text{ inches} \][/tex]
Therefore, one of the legs of the triangle measures approximately [tex]\(8.485 \text{ inches}\)[/tex].
