Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Discover a wealth of knowledge from experts across different disciplines on our comprehensive Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To determine which set of values could be the side lengths of a 30-60-90 triangle, we need to understand the properties of a 30-60-90 triangle. In such a triangle, the sides are in a specific ratio:
- The side opposite the 30-degree angle is the shortest and is denoted as \( x \).
- The side opposite the 60-degree angle is \( x \sqrt{3} \).
- The hypotenuse (opposite the 90-degree angle) is \( 2x \).
Given these properties, we will now check each set of values to see which one fits the \( 1 : \sqrt{3} : 2 \) ratio.
### Option A: \(\{6, 12, 12\sqrt{3}\}\)
1. Let's denote the shortest side as \( x = 6 \).
2. The side opposite the 60-degree angle should be \( 6 \sqrt{3} \). Here it is given as \( 12 \), which does not fit.
3. The hypotenuse should be \( 12 \), and here it is \( 12\sqrt{3} \), which does not fit either.
So, Option A is not correct.
### Option B: \(\{6, 6 \sqrt{3}, 12\}\)
1. Let the shortest side be \( x = 6 \).
2. The side opposite the 60-degree angle should be \( 6 \sqrt{3} \), which matches the given value \( 6 \sqrt{3} \).
3. The hypotenuse should be \( 12 \), and it is given as \( 12 \), which also matches.
So, Option B fits the required side lengths of a 30-60-90 triangle.
### Option C: \(\{6, 6 \sqrt{2}, 12\}\)
1. Let the shortest side be \( x = 6 \).
2. The side opposite the 60-degree angle should be \( 6 \sqrt{3} \), but here it is given as \( 6 \sqrt{2} \), which does not fit.
3. The hypotenuse should be \( 12 \), and it matches, but the middle side is incorrect.
So, Option C is not correct.
### Option D: \(\{6, 12, 12 \sqrt{2}\}\)
1. Let the shortest side be \( x = 6 \).
2. The side opposite the 60-degree angle should be \( 6 \sqrt{3} \), but here it is given as \( 12 \), which does not fit.
3. The hypotenuse should be \( 12 \), and it is \( 12 \sqrt{2} \), which also does not fit.
So, Option D is not correct.
### Conclusion
The only set of values that matches the side lengths of a 30-60-90 triangle is Option B: \(\{6, 6 \sqrt{3}, 12\}\).
So, the correct answer is:
B. [tex]\( \{6, 6 \sqrt{3}, 12\} \)[/tex]
- The side opposite the 30-degree angle is the shortest and is denoted as \( x \).
- The side opposite the 60-degree angle is \( x \sqrt{3} \).
- The hypotenuse (opposite the 90-degree angle) is \( 2x \).
Given these properties, we will now check each set of values to see which one fits the \( 1 : \sqrt{3} : 2 \) ratio.
### Option A: \(\{6, 12, 12\sqrt{3}\}\)
1. Let's denote the shortest side as \( x = 6 \).
2. The side opposite the 60-degree angle should be \( 6 \sqrt{3} \). Here it is given as \( 12 \), which does not fit.
3. The hypotenuse should be \( 12 \), and here it is \( 12\sqrt{3} \), which does not fit either.
So, Option A is not correct.
### Option B: \(\{6, 6 \sqrt{3}, 12\}\)
1. Let the shortest side be \( x = 6 \).
2. The side opposite the 60-degree angle should be \( 6 \sqrt{3} \), which matches the given value \( 6 \sqrt{3} \).
3. The hypotenuse should be \( 12 \), and it is given as \( 12 \), which also matches.
So, Option B fits the required side lengths of a 30-60-90 triangle.
### Option C: \(\{6, 6 \sqrt{2}, 12\}\)
1. Let the shortest side be \( x = 6 \).
2. The side opposite the 60-degree angle should be \( 6 \sqrt{3} \), but here it is given as \( 6 \sqrt{2} \), which does not fit.
3. The hypotenuse should be \( 12 \), and it matches, but the middle side is incorrect.
So, Option C is not correct.
### Option D: \(\{6, 12, 12 \sqrt{2}\}\)
1. Let the shortest side be \( x = 6 \).
2. The side opposite the 60-degree angle should be \( 6 \sqrt{3} \), but here it is given as \( 12 \), which does not fit.
3. The hypotenuse should be \( 12 \), and it is \( 12 \sqrt{2} \), which also does not fit.
So, Option D is not correct.
### Conclusion
The only set of values that matches the side lengths of a 30-60-90 triangle is Option B: \(\{6, 6 \sqrt{3}, 12\}\).
So, the correct answer is:
B. [tex]\( \{6, 6 \sqrt{3}, 12\} \)[/tex]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.