To determine which set of values could be the side lengths of a [tex]$30-60-90$[/tex] triangle, we should consider the fundamental properties of such a triangle.
A [tex]$30-60-90$[/tex] triangle has side lengths in the ratio [tex]\(1 : \sqrt{3} : 2\)[/tex]. This means:
- The shortest side (opposite the 30° angle) is [tex]\( x \)[/tex].
- The length of the side opposite the 60° angle is [tex]\( x \sqrt{3} \)[/tex].
- The hypotenuse (opposite the 90° angle) is [tex]\( 2x \)[/tex].
Using this property, we can examine each set of side lengths to determine if they adhere to this ratio.
Option A: [tex]\(\{4, 4 \sqrt{3}, 8 \sqrt{3}\}\)[/tex]
- Shortest side [tex]\( = 4 \)[/tex]
- Side opposite 60° angle [tex]\( = 4 \sqrt{3} \)[/tex]
- Hypotenuse [tex]\( = 8 \sqrt{3} \)[/tex]
Here, 8 [tex]\(\sqrt{3}\)[/tex] is not twice the shortest side 4, so this set does not maintain the [tex]\( 1 : \sqrt{3} : 2 \)[/tex] ratio.
Option B: [tex]\(\{4, 4 \sqrt{2}, 8 \sqrt{2}\}\)[/tex]
- Shortest side [tex]\( = 4 \)[/tex]
- Side opposite 60° angle [tex]\( = 4 \sqrt{2} \)[/tex]
- Hypotenuse [tex]\( = 8 \sqrt{2} \)[/tex]
This set does not follow the [tex]\( 1 : \sqrt{3} : 2 \)[/tex] ratio, as [tex]\(\sqrt{2}\)[/tex] is incorrect in place of [tex]\(\sqrt{3}\)[/tex].
Option C: [tex]\(\{4, 4 \sqrt{3}, 8\}\)[/tex]
- Shortest side [tex]\( = 4 \)[/tex]
- Side opposite 60° angle [tex]\( = 4 \sqrt{3} \)[/tex]
- Hypotenuse [tex]\( = 8 \)[/tex]
This set follows the [tex]\( 1 : \sqrt{3} : 2 \)[/tex] ratio because:
- [tex]\(4 \sqrt{3}\)[/tex] is consistent with [tex]\(4 \cdot \sqrt{3}\)[/tex], which is the correct length for the side opposite the 60° angle.
- The hypotenuse is [tex]\(2 \times 4 = 8\)[/tex], which fits the [tex]\( 2x \)[/tex] part of the ratio.
Option D: [tex]\(\{4, 4 \sqrt{2}, 8\}\)[/tex]
- Shortest side [tex]\( = 4 \)[/tex]
- Side opposite 60° angle [tex]\( = 4 \sqrt{2} \)[/tex]
- Hypotenuse [tex]\( = 8 \)[/tex]
This set does not follow the [tex]\( 1 : \sqrt{3} : 2 \)[/tex] ratio as [tex]\(\sqrt{2}\)[/tex] is incorrect.
The correct set of side lengths that match the properties of a [tex]$30-60-90$[/tex] triangle is:
[tex]\[
\boxed{3} \ (\{4, 4 \sqrt{3}, 8\})
\][/tex]
