Certainly! Let's discuss how to determine the extended ratio relating the side lengths of a 30-60-90 triangle.
A 30-60-90 triangle is a special type of right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees. The side lengths opposite these angles have a specific ratio.
1. The side opposite the 30-degree angle is the shortest side. Let's call this side [tex]\( x \)[/tex].
2. The side opposite the 60-degree angle is longer. The length of this side is [tex]\( x \sqrt{3} \)[/tex].
3. The side opposite the 90-degree angle, which is the hypotenuse, is the longest. The hypotenuse has a length of [tex]\( 2x \)[/tex].
Given this, the extended ratio of the side lengths in a 30-60-90 triangle is:
[tex]\[ x : x \sqrt{3} : 2x \][/tex]
So among the provided choices:
A. [tex]\( x: x: x \sqrt{3} \)[/tex]
B. [tex]\( x: x: x \sqrt{2} \)[/tex]
C. [tex]\( x: x \sqrt{2}: 3x \)[/tex]
D. [tex]\( x: x \sqrt{3}: 2x \)[/tex]
The correct answer is:
[tex]\[ \boxed{x: x \sqrt{3}: 2x} \][/tex]
Therefore, the extended ratio relating the side lengths of a 30-60-90 triangle is given by option D.
