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Sagot :
To determine which of the given ratios could be the ratio of the length of the longer leg of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle to the length of its hypotenuse, we need to recall that in a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle, the ratio of the sides opposite these angles is as follows:
- The side opposite the [tex]\(30^\circ\)[/tex] angle (the shorter leg) is [tex]\(a\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle (the longer leg) is [tex]\(\sqrt{3}a\)[/tex].
- The hypotenuse is [tex]\(2a\)[/tex].
Therefore, the ratio of the longer leg to the hypotenuse is:
[tex]\[ \frac{\text{Longer leg}}{\text{Hypotenuse}} = \frac{\sqrt{3}a}{2a} = \frac{\sqrt{3}}{2} \][/tex]
Now let's analyze each given option to see which ratios match [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Option A: [tex]\(2: 3\sqrt{3}\)[/tex]
[tex]\[ \frac{2}{3\sqrt{3}} = \frac{2}{3\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{9} \][/tex]
This is not equal to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Option B: [tex]\(1: \sqrt{2}\)[/tex]
[tex]\[ \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \][/tex]
This is not equal to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Option C: [tex]\(2: 2\sqrt{2}\)[/tex]
[tex]\[ \frac{2}{2\sqrt{2}} = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \][/tex]
This is not equal to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Option D: [tex]\(\sqrt{3}: 2\)[/tex]
[tex]\[ \frac{\sqrt{3}}{2} \][/tex]
This matches exactly [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Option E: [tex]\(3: 2\sqrt{3}\)[/tex]
[tex]\[ \frac{3}{2\sqrt{3}} = \frac{3}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2} \][/tex]
This matches exactly [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Option F: [tex]\(\sqrt{2}: \sqrt{3}\)[/tex]
[tex]\[ \frac{\sqrt{2}}{\sqrt{3}} = \sqrt{\frac{2}{3}} \][/tex]
This is not equal to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Therefore, the ratios that match [tex]\(\frac{\sqrt{3}}{2}\)[/tex] are Options D and E.
So, the correct options are:
- D. [tex]\(\sqrt{3}: 2\)[/tex]
- E. [tex]\(3: 2\sqrt{3}\)[/tex]
- The side opposite the [tex]\(30^\circ\)[/tex] angle (the shorter leg) is [tex]\(a\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle (the longer leg) is [tex]\(\sqrt{3}a\)[/tex].
- The hypotenuse is [tex]\(2a\)[/tex].
Therefore, the ratio of the longer leg to the hypotenuse is:
[tex]\[ \frac{\text{Longer leg}}{\text{Hypotenuse}} = \frac{\sqrt{3}a}{2a} = \frac{\sqrt{3}}{2} \][/tex]
Now let's analyze each given option to see which ratios match [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Option A: [tex]\(2: 3\sqrt{3}\)[/tex]
[tex]\[ \frac{2}{3\sqrt{3}} = \frac{2}{3\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{9} \][/tex]
This is not equal to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Option B: [tex]\(1: \sqrt{2}\)[/tex]
[tex]\[ \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \][/tex]
This is not equal to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Option C: [tex]\(2: 2\sqrt{2}\)[/tex]
[tex]\[ \frac{2}{2\sqrt{2}} = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \][/tex]
This is not equal to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Option D: [tex]\(\sqrt{3}: 2\)[/tex]
[tex]\[ \frac{\sqrt{3}}{2} \][/tex]
This matches exactly [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Option E: [tex]\(3: 2\sqrt{3}\)[/tex]
[tex]\[ \frac{3}{2\sqrt{3}} = \frac{3}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2} \][/tex]
This matches exactly [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Option F: [tex]\(\sqrt{2}: \sqrt{3}\)[/tex]
[tex]\[ \frac{\sqrt{2}}{\sqrt{3}} = \sqrt{\frac{2}{3}} \][/tex]
This is not equal to [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Therefore, the ratios that match [tex]\(\frac{\sqrt{3}}{2}\)[/tex] are Options D and E.
So, the correct options are:
- D. [tex]\(\sqrt{3}: 2\)[/tex]
- E. [tex]\(3: 2\sqrt{3}\)[/tex]
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