Answer :
To determine which set of values could be the side lengths of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, we need to understand the properties of this specific type of triangle.
A [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle has side lengths in a specific ratio:
- The side opposite the [tex]\(30^\circ\)[/tex] angle is the shortest side, let's denote it by [tex]\(a\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle is [tex]\(a\sqrt{3}\)[/tex].
- The hypotenuse (the side opposite the [tex]\(90^\circ\)[/tex] angle) is [tex]\(2a\)[/tex].
Now, let's check each set of values to see if they conform to this ratio.
Option A: [tex]\(\{5, 5 \sqrt{2}, 10\}\)[/tex]
- The smallest side is [tex]\(5\)[/tex].
- The hypotenuse is [tex]\(10\)[/tex].
- If we check the ratio:
- [tex]\(5\)[/tex] to [tex]\(10\)[/tex] (smallest side to hypotenuse) should be [tex]\(1:2\)[/tex], which is correct.
- [tex]\(5 \sqrt{2}\)[/tex] should be the middle side, but [tex]\(5 \sqrt{2} \approx 7.07\)[/tex].
- The middle side should be [tex]\(5 \sqrt{3} \approx 8.66\)[/tex] but [tex]\(5 \sqrt{2}\)[/tex] is about [tex]\(7.07\)[/tex], so this does not match the required ratio. This set is not a [tex]\(30-60-90\)[/tex] triangle.
Option B: [tex]\(\{5, 10, 10 \sqrt{2}\}\)[/tex]
- The smallest side is [tex]\(5\)[/tex].
- The hypotenuse here is [tex]\(10 \sqrt{2}\)[/tex].
- If we check the ratio:
- [tex]\(5\)[/tex] to [tex]\(10 \sqrt{2}\)[/tex] is [tex]\(1:2\sqrt{2}\)[/tex], which is incorrect because it should have been [tex]\(1:2\)[/tex].
- The middle side [tex]\(10\)[/tex] should be [tex]\(5 \sqrt{3} \approx 8.66\)[/tex], which does not fit in the expected ratio. This set is not a [tex]\(30-60-90\)[/tex] triangle.
Option C: [tex]\(\{5, 5 \sqrt{3}, 10\}\)[/tex]
- The smallest side is [tex]\(5\)[/tex].
- The middle side is [tex]\(5 \sqrt{3}\)[/tex].
- The hypotenuse is [tex]\(10\)[/tex].
- If we check the ratio:
- [tex]\(5\)[/tex] to [tex]\(10\)[/tex] is [tex]\(1:2\)[/tex], which is correct.
- [tex]\(5 \sqrt{3}\)[/tex] to [tex]\(10\)[/tex] is [tex]\(\sqrt{3}/2\)[/tex], which is the required ratio.
- This set fits the required ratio of side lengths of a [tex]\(30-60-90\)[/tex] triangle, so it is correct.
Option D: [tex]\(\{5, 10, 10.5\}\)[/tex]
- The smallest side is [tex]\(5\)[/tex].
- The hypotenuse is [tex]\(10.5\)[/tex].
- If we check the ratio:
- [tex]\(5\)[/tex] to [tex]\(10.5\)[/tex] is around [tex]\(1:2.1\)[/tex], which is incorrect because it should have been [tex]\(1:2\)[/tex].
- The middle side [tex]\(10\)[/tex] should be [tex]\(5 \sqrt{3} \approx 8.66\)[/tex], which does not fit in the expected ratio.
Therefore, the set of values that could be the side lengths of a [tex]\(30-60-90\)[/tex] triangle is:
Option C: [tex]\(\{5, 5 \sqrt{3}, 10\}\)[/tex].
A [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle has side lengths in a specific ratio:
- The side opposite the [tex]\(30^\circ\)[/tex] angle is the shortest side, let's denote it by [tex]\(a\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle is [tex]\(a\sqrt{3}\)[/tex].
- The hypotenuse (the side opposite the [tex]\(90^\circ\)[/tex] angle) is [tex]\(2a\)[/tex].
Now, let's check each set of values to see if they conform to this ratio.
Option A: [tex]\(\{5, 5 \sqrt{2}, 10\}\)[/tex]
- The smallest side is [tex]\(5\)[/tex].
- The hypotenuse is [tex]\(10\)[/tex].
- If we check the ratio:
- [tex]\(5\)[/tex] to [tex]\(10\)[/tex] (smallest side to hypotenuse) should be [tex]\(1:2\)[/tex], which is correct.
- [tex]\(5 \sqrt{2}\)[/tex] should be the middle side, but [tex]\(5 \sqrt{2} \approx 7.07\)[/tex].
- The middle side should be [tex]\(5 \sqrt{3} \approx 8.66\)[/tex] but [tex]\(5 \sqrt{2}\)[/tex] is about [tex]\(7.07\)[/tex], so this does not match the required ratio. This set is not a [tex]\(30-60-90\)[/tex] triangle.
Option B: [tex]\(\{5, 10, 10 \sqrt{2}\}\)[/tex]
- The smallest side is [tex]\(5\)[/tex].
- The hypotenuse here is [tex]\(10 \sqrt{2}\)[/tex].
- If we check the ratio:
- [tex]\(5\)[/tex] to [tex]\(10 \sqrt{2}\)[/tex] is [tex]\(1:2\sqrt{2}\)[/tex], which is incorrect because it should have been [tex]\(1:2\)[/tex].
- The middle side [tex]\(10\)[/tex] should be [tex]\(5 \sqrt{3} \approx 8.66\)[/tex], which does not fit in the expected ratio. This set is not a [tex]\(30-60-90\)[/tex] triangle.
Option C: [tex]\(\{5, 5 \sqrt{3}, 10\}\)[/tex]
- The smallest side is [tex]\(5\)[/tex].
- The middle side is [tex]\(5 \sqrt{3}\)[/tex].
- The hypotenuse is [tex]\(10\)[/tex].
- If we check the ratio:
- [tex]\(5\)[/tex] to [tex]\(10\)[/tex] is [tex]\(1:2\)[/tex], which is correct.
- [tex]\(5 \sqrt{3}\)[/tex] to [tex]\(10\)[/tex] is [tex]\(\sqrt{3}/2\)[/tex], which is the required ratio.
- This set fits the required ratio of side lengths of a [tex]\(30-60-90\)[/tex] triangle, so it is correct.
Option D: [tex]\(\{5, 10, 10.5\}\)[/tex]
- The smallest side is [tex]\(5\)[/tex].
- The hypotenuse is [tex]\(10.5\)[/tex].
- If we check the ratio:
- [tex]\(5\)[/tex] to [tex]\(10.5\)[/tex] is around [tex]\(1:2.1\)[/tex], which is incorrect because it should have been [tex]\(1:2\)[/tex].
- The middle side [tex]\(10\)[/tex] should be [tex]\(5 \sqrt{3} \approx 8.66\)[/tex], which does not fit in the expected ratio.
Therefore, the set of values that could be the side lengths of a [tex]\(30-60-90\)[/tex] triangle is:
Option C: [tex]\(\{5, 5 \sqrt{3}, 10\}\)[/tex].
