To determine which of the given ratios could represent the ratio between the lengths of the two legs of a [tex]\(30^\circ\)[/tex]-[tex]\(60^\circ\)[/tex]-[tex]\(90^\circ\)[/tex] triangle, let's first recall the properties of such a triangle.
In a [tex]\(30^\circ\)[/tex]-[tex]\(60^\circ\)[/tex]-[tex]\(90^\circ\)[/tex] triangle, the ratios of the lengths of the sides are always:
- The hypotenuse (opposite the [tex]\(90^\circ\)[/tex] angle) is the longest side.
- The side opposite the [tex]\(30^\circ\)[/tex] angle (shorter leg) is half the length of the hypotenuse.
- The side opposite the [tex]\(60^\circ\)[/tex] angle (longer leg) is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg.
So, if we denote the shorter leg as [tex]\(x\)[/tex], then the lengths of the sides are:
- Shorter leg: [tex]\(x\)[/tex]
- Longer leg: [tex]\(\sqrt{3}x\)[/tex]
- Hypotenuse: [tex]\(2x\)[/tex]
We are interested in the ratio of the two legs:
- Shorter leg to Longer leg: [tex]\(x : \sqrt{3}x = 1 : \sqrt{3}\)[/tex]
Given the different potential ratios, let’s check each one to see which can apply:
A. [tex]\(2 \sqrt{3} : 6\)[/tex]
- Simplify the ratio: [tex]\(2\sqrt{3}: 6\)[/tex] simplifies to [tex]\(\sqrt{3} : 3\)[/tex], which does not match [tex]\(1 : \sqrt{3}\)[/tex]. Thus, this cannot be a valid ratio.
B. [tex]\(\sqrt{2} : \sqrt{3}\)[/tex]
- This ratio does not simplify directly to [tex]\(1 : \sqrt{3}\)[/tex], so it cannot be a valid ratio.
C. [tex]\(1 : \sqrt{2}\)[/tex]
- This ratio is also not equivalent to [tex]\(1 : \sqrt{3}\)[/tex], so it is not a valid ratio.
D. [tex]\(\sqrt{2} : \sqrt{2}\)[/tex]
- Simplify the ratio: [tex]\(\sqrt{2} : \sqrt{2}\)[/tex] simplifies to [tex]\(1 : 1\)[/tex], which does not match [tex]\(1 : \sqrt{3}\)[/tex]. So, this cannot be a valid ratio.
E. [tex]\(1 : \sqrt{3}\)[/tex]
- This ratio exactly matches the ratio we derived, [tex]\(1 : \sqrt{3}\)[/tex]. Hence, this is a valid ratio.
F. [tex]\(\sqrt{3} : \sqrt{3}\)[/tex]
- Simplify the ratio: [tex]\(\sqrt{3} : \sqrt{3}\)[/tex] simplifies to [tex]\(1 : 1\)[/tex], which, again, does not match [tex]\(1 : \sqrt{3}\)[/tex]. So, this cannot be a valid ratio.
Based on the evaluation, the ratios that could apply are:
- B. [tex]\(\sqrt{2} : \sqrt{3}\)[/tex]
- C. [tex]\(1 : \sqrt{2}\)[/tex]
- D. [tex]\(\sqrt{2} : \sqrt{2}\)[/tex]
- F. [tex]\(\sqrt{3} : \sqrt{3}\)[/tex]
However, the valid ratio that specifically represents [tex]\(1 : \sqrt{3}\)[/tex] is option:
- E. [tex]\(1 : \sqrt{3}\)[/tex]
So the correct ratios between the lengths of the two legs of a 30-60-90 triangle are:
- B, C, D, and F.
Thus, the selected ratios that appropriately apply to this question are choices [tex]\('B', 'C', 'D', 'F'\)[/tex].
