To determine the ratio of the length of one leg to the length of the other leg in a 45-45-90 right triangle, we start by understanding the properties of this special type of triangle.
A 45-45-90 triangle is a right triangle where the two non-right angles are both 45 degrees. This is one of the special triangles, and it has the following properties:
1. It is an isosceles right triangle, meaning the legs opposite the 45-degree angles are congruent (equal in length).
2. The hypotenuse (the side opposite the right angle) is [tex]\( \sqrt{2} \)[/tex] times the length of each leg.
Since both legs of the triangle are equal, we denote the length of each leg as [tex]\( x \)[/tex]. Thus, the leg lengths are equal to [tex]\( x \)[/tex].
Given that the legs are congruent, the ratio of the length of one leg to the length of the other leg is simply:
[tex]\[
\frac{x}{x} = 1
\][/tex]
Thus, the ratio of the length of one leg to the length of the other leg in a 45-45-90 right triangle is 1:1.
Therefore, the correct answer is:
[tex]\[ \boxed{1:1} \][/tex]
