In a 45-45-90 right triangle, what is the ratio of the length of one leg to the length of the other leg?

A. [tex]\(\sqrt{2}: 1\)[/tex]
B. [tex]\(1: 1\)[/tex]
C. [tex]\(1: \sqrt{2}\)[/tex]
D. [tex]\(2: 1\)[/tex]



Answer :

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]