To solve this problem, we need to determine the correct ratio of the length of the longer leg of a [tex]$30^\circ$[/tex]-[tex]$60^\circ$[/tex]-[tex]$90^\circ$[/tex] triangle to the length of its hypotenuse. Here's the step-by-step approach:
1. Understand the properties of a [tex]$30^\circ$[/tex]-[tex]$60^\circ$[/tex]-[tex]$90^\circ$[/tex] triangle:
- The sides of a [tex]$30^\circ$[/tex]-[tex]$60^\circ$[/tex]-[tex]$90^\circ$[/tex] triangle are in the ratio [tex]\( 1 : \sqrt{3} : 2 \)[/tex].
- Specifically, if the shorter leg (opposite the [tex]$30^\circ$[/tex] angle) is [tex]\( x \)[/tex]:
- The longer leg (opposite the [tex]$60^\circ$[/tex] angle) is [tex]\( x \sqrt{3} \)[/tex].
- The hypotenuse (opposite the [tex]$90^\circ$[/tex] angle) is [tex]\( 2x \)[/tex].
2. Determine the ratio of the longer leg to the hypotenuse:
- The longer leg is [tex]\( x \sqrt{3} \)[/tex] and the hypotenuse is [tex]\( 2x \)[/tex].
- Therefore, the ratio of the longer leg to the hypotenuse is: [tex]\( \frac{x \sqrt{3}}{2x} = \frac{\sqrt{3}}{2} \)[/tex].
Hence, we are looking for the ratio [tex]\( \sqrt{3} : 2 \)[/tex] among the provided options.
3. Evaluate each given ratio:
- Option A [tex]\( \sqrt{3}: 2 \)[/tex]: This is exactly the value we are looking for.
- Option B [tex]\( 2: 3 \sqrt{5} \)[/tex]: This is not the same as [tex]\( \sqrt{3}: 2 \)[/tex].
- Option C [tex]\( 1: \sqrt{2} \)[/tex]: This is not the same as [tex]\( \sqrt{3}: 2 \)[/tex].
- Option D [tex]\( \sqrt{2}: \sqrt{3} \)[/tex]: This is not the same as [tex]\( \sqrt{3}: 2 \)[/tex].
- Option E [tex]\( 2: 2 \sqrt{2} \)[/tex]: This can be simplified to [tex]\( 1: \sqrt{2} \)[/tex], but this is not the same as [tex]\( \sqrt{3}: 2 \)[/tex].
- Option F [tex]\( 3: 2 \sqrt{3} \)[/tex]: This is not the same as [tex]\( \sqrt{3}: 2 \)[/tex].
4. Validate our options:
- The only option that matches [tex]\( \sqrt{3} : 2 \)[/tex] is Option A.
Therefore, the correct ratio of the length of the longer leg of a [tex]$30^\circ$[/tex]-[tex]$60^\circ$[/tex]-[tex]$90^\circ$[/tex] triangle to the length of its hypotenuse is given by:
A. [tex]\(\sqrt{3} : 2\)[/tex]
