Answer :
To determine which set of values could be the side lengths of a [tex]$30$[/tex]-[tex]$60$[/tex]-[tex]$90$[/tex] triangle, we need to recognize the characteristic ratios of the side lengths for such a triangle. In a [tex]$30$[/tex]-[tex]$60$[/tex]-[tex]$90$[/tex] triangle, the side lengths are in the ratio [tex]\(1 : \sqrt{3} : 2\)[/tex].
Now let's evaluate each given set of values:
### A. [tex]\(\{4, 4 \sqrt{3}, 8 \sqrt{3}\}\)[/tex]
First, we consider the side lengths [tex]\(4, 4 \sqrt{3}, 8 \sqrt{3}\)[/tex]. Let's find their ratios relative to the first side length (which is [tex]\(4\)[/tex]):
- Ratio of the second side to the first side: [tex]\(\frac{4 \sqrt{3}}{4} = \sqrt{3}\)[/tex]
- Ratio of the third side to the first side: [tex]\(\frac{8 \sqrt{3}}{4} = 2 \sqrt{3}\)[/tex]
So, the ratios for set A are [tex]\(\{1, \sqrt{3}, 2\sqrt{3}\}\)[/tex]. This does not match the expected ratio [tex]\(1 : \sqrt{3} : 2 \)[/tex].
### B. [tex]\(\{4, 4 \sqrt{2}, 8 \sqrt{2}\}\)[/tex]
Now we consider the side lengths [tex]\(4, 4 \sqrt{2}, 8 \sqrt{2}\)[/tex]. Let's find their ratios relative to the first side length (which is [tex]\(4\)[/tex]):
- Ratio of the second side to the first side: [tex]\(\frac{4 \sqrt{2}}{4} = \sqrt{2}\)[/tex]
- Ratio of the third side to the first side: [tex]\(\frac{8 \sqrt{2}}{4} = 2 \sqrt{2}\)[/tex]
So, the ratios for set B are [tex]\(\{1, \sqrt{2}, 2\sqrt{2}\}\)[/tex]. This does not match the expected ratio [tex]\(1 : \sqrt{3} : 2 \)[/tex].
### C. [tex]\(\{4, 4 \sqrt{3}, 8\}\)[/tex]
Next, we consider the side lengths [tex]\(4, 4 \sqrt{3}, 8\)[/tex]. Let's find their ratios relative to the first side length (which is [tex]\(4\)[/tex]):
- Ratio of the second side to the first side: [tex]\(\frac{4 \sqrt{3}}{4} = \sqrt{3}\)[/tex]
- Ratio of the third side to the first side: [tex]\(\frac{8}{4} = 2\)[/tex]
So, the ratios for set C are [tex]\(\{1, \sqrt{3}, 2\}\)[/tex]. This matches the expected ratio [tex]\(1 : \sqrt{3} : 2 \)[/tex].
### D. [tex]\(\{4, 4 \sqrt{2}, 8\}\)[/tex]
Finally, we consider the side lengths [tex]\(4, 4 \sqrt{2}, 8\)[/tex]. Let's find their ratios relative to the first side length (which is [tex]\(4\)[/tex]):
- Ratio of the second side to the first side: [tex]\(\frac{4 \sqrt{2}}{4} = \sqrt{2}\)[/tex]
- Ratio of the third side to the first side: [tex]\(\frac{8}{4} = 2\)[/tex]
So, the ratios for set D are [tex]\(\{1, \sqrt{2}, 2\}\)[/tex]. This does not match the expected ratio [tex]\(1 : \sqrt{3} : 2 \)[/tex].
Hence, the correct set that could be the side lengths of a [tex]$30$[/tex]-[tex]$60$[/tex]-[tex]$90$[/tex] triangle is [tex]\( \text{C.} \{4, 4 \sqrt{3}, 8\} \)[/tex].
Now let's evaluate each given set of values:
### A. [tex]\(\{4, 4 \sqrt{3}, 8 \sqrt{3}\}\)[/tex]
First, we consider the side lengths [tex]\(4, 4 \sqrt{3}, 8 \sqrt{3}\)[/tex]. Let's find their ratios relative to the first side length (which is [tex]\(4\)[/tex]):
- Ratio of the second side to the first side: [tex]\(\frac{4 \sqrt{3}}{4} = \sqrt{3}\)[/tex]
- Ratio of the third side to the first side: [tex]\(\frac{8 \sqrt{3}}{4} = 2 \sqrt{3}\)[/tex]
So, the ratios for set A are [tex]\(\{1, \sqrt{3}, 2\sqrt{3}\}\)[/tex]. This does not match the expected ratio [tex]\(1 : \sqrt{3} : 2 \)[/tex].
### B. [tex]\(\{4, 4 \sqrt{2}, 8 \sqrt{2}\}\)[/tex]
Now we consider the side lengths [tex]\(4, 4 \sqrt{2}, 8 \sqrt{2}\)[/tex]. Let's find their ratios relative to the first side length (which is [tex]\(4\)[/tex]):
- Ratio of the second side to the first side: [tex]\(\frac{4 \sqrt{2}}{4} = \sqrt{2}\)[/tex]
- Ratio of the third side to the first side: [tex]\(\frac{8 \sqrt{2}}{4} = 2 \sqrt{2}\)[/tex]
So, the ratios for set B are [tex]\(\{1, \sqrt{2}, 2\sqrt{2}\}\)[/tex]. This does not match the expected ratio [tex]\(1 : \sqrt{3} : 2 \)[/tex].
### C. [tex]\(\{4, 4 \sqrt{3}, 8\}\)[/tex]
Next, we consider the side lengths [tex]\(4, 4 \sqrt{3}, 8\)[/tex]. Let's find their ratios relative to the first side length (which is [tex]\(4\)[/tex]):
- Ratio of the second side to the first side: [tex]\(\frac{4 \sqrt{3}}{4} = \sqrt{3}\)[/tex]
- Ratio of the third side to the first side: [tex]\(\frac{8}{4} = 2\)[/tex]
So, the ratios for set C are [tex]\(\{1, \sqrt{3}, 2\}\)[/tex]. This matches the expected ratio [tex]\(1 : \sqrt{3} : 2 \)[/tex].
### D. [tex]\(\{4, 4 \sqrt{2}, 8\}\)[/tex]
Finally, we consider the side lengths [tex]\(4, 4 \sqrt{2}, 8\)[/tex]. Let's find their ratios relative to the first side length (which is [tex]\(4\)[/tex]):
- Ratio of the second side to the first side: [tex]\(\frac{4 \sqrt{2}}{4} = \sqrt{2}\)[/tex]
- Ratio of the third side to the first side: [tex]\(\frac{8}{4} = 2\)[/tex]
So, the ratios for set D are [tex]\(\{1, \sqrt{2}, 2\}\)[/tex]. This does not match the expected ratio [tex]\(1 : \sqrt{3} : 2 \)[/tex].
Hence, the correct set that could be the side lengths of a [tex]$30$[/tex]-[tex]$60$[/tex]-[tex]$90$[/tex] triangle is [tex]\( \text{C.} \{4, 4 \sqrt{3}, 8\} \)[/tex].
