To determine the extended ratio relating the side lengths of a 45-45-90 triangle, let's first recall some fundamental properties of this type of triangle.
A 45-45-90 triangle is a special type of right triangle where the two legs adjacent to the right angle are of equal length, and the hypotenuse is opposite the right angle. Due to its geometric properties:
1. The two legs (let's call the length of each leg [tex]\( x \)[/tex]) will always be equal in length.
2. The hypotenuse will be [tex]\( x \sqrt{2} \)[/tex].
Thus, the extended ratio of the side lengths of a 45-45-90 triangle should include:
- The two equal legs are both [tex]\( x \)[/tex].
- The hypotenuse, which is [tex]\( x \sqrt{2} \)[/tex].
Let's examine the given options:
A. [tex]\( x: x \sqrt{3}: 2 x \)[/tex]
- This ratio does not represent the correct relationship as it involves [tex]\( \sqrt{3} \)[/tex] and 2, which are not relevant to a 45-45-90 triangle.
B. [tex]\( x: x: x \sqrt{2} \)[/tex]
- This ratio accurately depicts the relationship of the side lengths of a 45-45-90 triangle. The two legs are both [tex]\( x \)[/tex], and the hypotenuse is [tex]\( x \sqrt{2} \)[/tex].
C. [tex]\( x: x \sqrt{2}: 3 x \)[/tex]
- This ratio is incorrect as it introduces [tex]\( 3 x \)[/tex], which does not correspond to the properties of the triangle in question.
D. [tex]\( x: x: x \sqrt{3} \)[/tex]
- This ratio also is incorrect as [tex]\( \sqrt{3} \)[/tex] does not appear in the relationship of the side lengths of a 45-45-90 triangle.
Thus, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
