A 30-60-90 triangle is a special type of right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees. The sides of a 30-60-90 triangle always have the same ratio, which is 1 : √3 : 2.
This means that if the shortest side (opposite the 30-degree angle) has length 'a', then:
- The side opposite the 60-degree angle (the longer leg) will be 'a√3'.
- The side opposite the 90-degree angle (the hypotenuse) will be '2a'.
Let's check each of the options:
A. (5, 5√2, 10): This does not follow the 1 : √3 : 2 ratio.
B. (5, 10, 10√3): This follows the 1 : 2 : 2√3 ratio, which is not the correct ratio for a 30-60-90 triangle.
C. (5, 10, 10^2): This does not follow the 1 : √3 : 2 ratio.
D. (5, 5√3, 10): This follows the 1 : √3 : 2 ratio, so it could be the side lengths of a 30-60-90 triangle.
So, the correct answer is option D. (5, 5√3, 10).