To determine the length of one leg of a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle when the hypotenuse is [tex]\( 10 \sqrt{5} \)[/tex] inches, we can use the properties of this special type of triangle. In a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle, the legs are congruent, and the relationship between the legs (denoted as [tex]\( a \)[/tex]) and the hypotenuse (denoted as [tex]\( c \)[/tex]) is given by
[tex]\[ c = a \sqrt{2} \][/tex]
Given that the hypotenuse [tex]\( c \)[/tex] is [tex]\( 10 \sqrt{5} \)[/tex], we set up the equation
[tex]\[ 10 \sqrt{5} = a \sqrt{2} \][/tex]
To solve for [tex]\( a \)[/tex], the length of one leg, we divide both sides of the equation by [tex]\( \sqrt{2} \)[/tex],
[tex]\[ a = \frac{10 \sqrt{5}}{\sqrt{2}} \][/tex]
Next, we rationalize the denominator by multiplying both the numerator and the denominator by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ a = \frac{10 \sqrt{5} \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{10 \sqrt{10}}{2} = 5 \sqrt{10} \][/tex]
Thus, the length of one leg of the triangle is
[tex]\[ \boxed{5 \sqrt{10}} \][/tex]
Therefore, the correct answer is [tex]\( 5 \sqrt{10} \)[/tex].
