In a 45-45-90 right triangle, the two legs are congruent, meaning they have the same length. This happens because the two non-right angles in this type of triangle are both 45 degrees, making the triangle isosceles with the right angle as the vertex angle.
Let's denote the length of one leg as [tex]\( a \)[/tex] and the length of the other leg as [tex]\( b \)[/tex]. Since both legs are of equal length, we have:
[tex]\[ a = b \][/tex]
To express the ratio of the length of one leg to the length of the other leg, we simply take the ratio of [tex]\( a \)[/tex] to [tex]\( b \)[/tex]:
[tex]\[ \frac{a}{b} \][/tex]
Given that [tex]\( a = b \)[/tex], this ratio simplifies as:
[tex]\[ \frac{a}{a} = 1 \][/tex]
Therefore, the ratio of the length of one leg to the length of the other leg in a 45-45-90 right triangle is:
[tex]\[ 1: 1 \][/tex]
Hence, the correct answer is:
C. [tex]\( 1: 1 \)[/tex]
