Which of the following could be the ratio between the lengths of the legs of a 30-60-90 triangle? Check all that apply: √3:3 1:√3

In a 30-60-90 triangle, the lengths of the sides are in a known ratio. The side opposite the 30° angle is the shortest, and we'll call this length 'x'. The side opposite the 60° angle is 'x√3', and the hypotenuse, which is the side opposite the 90° angle, is '2x'. Now, let's examine the given options:

√3:3 - This cannot be the ratio between the legs of a 30-60-90 triangle as described above because in the standard ratio, the two non-hypotenuse sides should be 'x' and 'x√3'. If we take the square root of 3 (√3) for 'x', the other leg should be √3 times √3, which equals 3. However, the hypotenuse should be 2 times √3, which is not present in this ratio. So, √3:3 does not represent the ratio between the legs.

1:√3 - This could represent the ratio between the lengths of the legs of a 30-60-90 triangle. Here, if we consider 'x' as 1, then the length of the side opposite the 60° angle would be 'x√3', which means '1√3' or just '√3'. This ratio correctly represents the proportion between the lengths of the legs of a 30-60-90 triangle.

So, the ratio that applies is 1:√3.

Extra: Understanding the 30-60-90 triangle is helpful in trigonometry and geometry. As mentioned earlier, the lengths of the sides of such a triangle are always in the ratio 1:√3:2. This special triangle is half of an equilateral triangle, which is why the ratios are consistent. The short leg (opposite the 30° angle) forms the base of the equilateral triangle, the longer leg (opposite the 60° angle) forms the height, and the hypotenuse is equal to the side of the equilateral triangle. You can often use these properties to solve problems involving right triangles without the need for additional calculations or trigonometric functions.