Answer :
To determine which set of values could be the side lengths of a 30-60-90 triangle, we need to recall the special properties of such triangles. In a 30-60-90 triangle, the sides are in a specific ratio:
- The shortest side (opposite the 30-degree angle) is [tex]\(a\)[/tex].
- The side opposite the 60-degree angle is [tex]\(a\sqrt{3}\)[/tex].
- The hypotenuse (opposite the 90-degree angle) is [tex]\(2a\)[/tex].
Given this, let's analyze each set of side lengths to see which one fits this ratio.
### Option A: [tex]\(\{5, 10, 10\sqrt{3}\}\)[/tex]
1. Let's designate [tex]\(5\)[/tex] as [tex]\(a\)[/tex], the shortest side.
2. Then the hypotenuse should be [tex]\(2a = 2 \times 5 = 10\)[/tex].
3. The side opposite the 60-degree angle should be [tex]\(a\sqrt{3} = 5\sqrt{3}\)[/tex].
Let's check the ratios:
- The ratio of the hypotenuse to the shortest side is [tex]\( \frac{10}{5} = 2\)[/tex].
- The ratio of the side opposite the 60-degree angle to the shortest side: [tex]\( \frac{10\sqrt{3}}{5} = 2\sqrt{3}\)[/tex].
So, option A fits the 30-60-90 triangle ratio since these calculated values match the expected ratios: [tex]\(2.0 \)[/tex] for the hypotenuse and approximately [tex]\(3.464\)[/tex] for the longer leg.
### Option B: [tex]\(\{5, 5\sqrt{2}, 10\}\)[/tex]
1. Let's designate [tex]\(5\)[/tex] as [tex]\(a\)[/tex], the shortest side.
2. Then the hypotenuse should be [tex]\(2a = 2 \times 5 = 10\)[/tex].
3. The side opposite the 60-degree angle should be [tex]\(a\sqrt{3} = 5\sqrt{3}\)[/tex].
Checking these given values against the 30-60-90 triangle ratios:
- The side opposite the 60-degree angle here is [tex]\(5\sqrt{2}\)[/tex], not [tex]\(5\sqrt{3}\)[/tex].
This does not match the expected side ratios of a 30-60-90 triangle.
### Option C: [tex]\(\{5, 5\sqrt{3}, 10\}\)[/tex]
1. Let's designate [tex]\(5\)[/tex] as [tex]\(a\)[/tex], the shortest side.
2. Then the hypotenuse should be [tex]\(2a = 2 \times 5 = 10\)[/tex].
3. The side opposite the 60-degree angle should be [tex]\(a\sqrt{3} = 5\sqrt{3}\)[/tex].
Checking the ratios:
- The ratio of the hypotenuse to the shortest side is [tex]\( \frac{10}{5} = 2\)[/tex].
- The ratio of the side opposite the 60-degree angle to the shortest side: [tex]\( \frac{5\sqrt{3}}{5} = \sqrt{3}\)[/tex].
These values fit perfectly with the expected ratios of [tex]\(2.0\)[/tex] for the hypotenuse and approximately [tex]\(1.732\)[/tex] for the longer leg.
### Option D: [tex]\(\{5, 10, 10\sqrt{2}\}\)[/tex]
1. Let's designate [tex]\(5\)[/tex] as [tex]\(a\)[/tex], the shortest side.
2. Then the hypotenuse should be [tex]\(2a = 2 \times 5 = 10\)[/tex].
3. The side opposite the 60-degree angle should be [tex]\(a\sqrt{3} = 5\sqrt{3}\)[/tex].
Checking these given values against the 30-60-90 triangle ratios:
- The side opposite the 60-degree angle here is [tex]\(10\sqrt{2}\)[/tex], which does not match [tex]\(5\sqrt{3}\)[/tex].
This does not match the expected side ratios of a 30-60-90 triangle.
### Conclusion
The correct set of values that could be the side lengths of a 30-60-90 triangle is:
[tex]\[ \boxed{\{5,10,10\sqrt{3}\}} \][/tex]
- The shortest side (opposite the 30-degree angle) is [tex]\(a\)[/tex].
- The side opposite the 60-degree angle is [tex]\(a\sqrt{3}\)[/tex].
- The hypotenuse (opposite the 90-degree angle) is [tex]\(2a\)[/tex].
Given this, let's analyze each set of side lengths to see which one fits this ratio.
### Option A: [tex]\(\{5, 10, 10\sqrt{3}\}\)[/tex]
1. Let's designate [tex]\(5\)[/tex] as [tex]\(a\)[/tex], the shortest side.
2. Then the hypotenuse should be [tex]\(2a = 2 \times 5 = 10\)[/tex].
3. The side opposite the 60-degree angle should be [tex]\(a\sqrt{3} = 5\sqrt{3}\)[/tex].
Let's check the ratios:
- The ratio of the hypotenuse to the shortest side is [tex]\( \frac{10}{5} = 2\)[/tex].
- The ratio of the side opposite the 60-degree angle to the shortest side: [tex]\( \frac{10\sqrt{3}}{5} = 2\sqrt{3}\)[/tex].
So, option A fits the 30-60-90 triangle ratio since these calculated values match the expected ratios: [tex]\(2.0 \)[/tex] for the hypotenuse and approximately [tex]\(3.464\)[/tex] for the longer leg.
### Option B: [tex]\(\{5, 5\sqrt{2}, 10\}\)[/tex]
1. Let's designate [tex]\(5\)[/tex] as [tex]\(a\)[/tex], the shortest side.
2. Then the hypotenuse should be [tex]\(2a = 2 \times 5 = 10\)[/tex].
3. The side opposite the 60-degree angle should be [tex]\(a\sqrt{3} = 5\sqrt{3}\)[/tex].
Checking these given values against the 30-60-90 triangle ratios:
- The side opposite the 60-degree angle here is [tex]\(5\sqrt{2}\)[/tex], not [tex]\(5\sqrt{3}\)[/tex].
This does not match the expected side ratios of a 30-60-90 triangle.
### Option C: [tex]\(\{5, 5\sqrt{3}, 10\}\)[/tex]
1. Let's designate [tex]\(5\)[/tex] as [tex]\(a\)[/tex], the shortest side.
2. Then the hypotenuse should be [tex]\(2a = 2 \times 5 = 10\)[/tex].
3. The side opposite the 60-degree angle should be [tex]\(a\sqrt{3} = 5\sqrt{3}\)[/tex].
Checking the ratios:
- The ratio of the hypotenuse to the shortest side is [tex]\( \frac{10}{5} = 2\)[/tex].
- The ratio of the side opposite the 60-degree angle to the shortest side: [tex]\( \frac{5\sqrt{3}}{5} = \sqrt{3}\)[/tex].
These values fit perfectly with the expected ratios of [tex]\(2.0\)[/tex] for the hypotenuse and approximately [tex]\(1.732\)[/tex] for the longer leg.
### Option D: [tex]\(\{5, 10, 10\sqrt{2}\}\)[/tex]
1. Let's designate [tex]\(5\)[/tex] as [tex]\(a\)[/tex], the shortest side.
2. Then the hypotenuse should be [tex]\(2a = 2 \times 5 = 10\)[/tex].
3. The side opposite the 60-degree angle should be [tex]\(a\sqrt{3} = 5\sqrt{3}\)[/tex].
Checking these given values against the 30-60-90 triangle ratios:
- The side opposite the 60-degree angle here is [tex]\(10\sqrt{2}\)[/tex], which does not match [tex]\(5\sqrt{3}\)[/tex].
This does not match the expected side ratios of a 30-60-90 triangle.
### Conclusion
The correct set of values that could be the side lengths of a 30-60-90 triangle is:
[tex]\[ \boxed{\{5,10,10\sqrt{3}\}} \][/tex]
