Discover how IDNLearn.com can help you find the answers you need quickly and easily. Discover detailed and accurate answers to your questions from our knowledgeable and dedicated community members.
Sagot :
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].
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. IDNLearn.com is your source for precise answers. Thank you for visiting, and we look forward to helping you again soon.
