Answer :
A \(30-60-90\) triangle is a special type of right triangle where the angles are \(30^\circ\), \(60^\circ\), and \(90^\circ\). The sides of such a triangle have a specific ratio:
- The shortest side (opposite the \(30^\circ\) angle) is \(x\).
- The longer leg (opposite the \(60^\circ\) angle) is \(x\sqrt{3}\).
- The hypotenuse (opposite the \(90^\circ\) angle) is \(2x\).
In this problem, we are asked to find the ratio of the length of the longer leg to the length of the hypotenuse. The ratio for a \(30-60-90\) triangle is known to be:
[tex]\[ \text{Longer Leg : Hypotenuse} = x\sqrt{3} : 2x = \sqrt{3} : 2 \][/tex]
We need to verify which of the given options match this ratio.
### Option A: \(\sqrt{3} : 2\)
This matches our ratio exactly.
### Option B: \(\sqrt{3} : \sqrt{3}\)
Here, both parts of the ratio are equal, so it simplifies to \(1 : 1\), which does not match \(\sqrt{3} : 2\).
### Option C: \(2\sqrt{3} : 4\)
Let's simplify this ratio:
[tex]\[ \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \][/tex]
This matches \(\sqrt{3} : 2\).
### Option D: \(2 : 2\sqrt{2}\)
Let's simplify this ratio:
[tex]\[ \frac{2}{2\sqrt{2}} = \frac{2}{2\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \][/tex]
This does not match \(\sqrt{3} : 2\).
### Option E: \(\sqrt{2} : \sqrt{3}\)
This ratio cannot be simplified to match \(\sqrt{3} : 2\).
### Option F: \(1 : \sqrt{3}\)
This ratio can be simplified as follows:
[tex]\[ \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \][/tex]
This does not match \(\sqrt{3} : 2\).
Based on these evaluations, the ratios that match the ratio of the length of the longer leg to the length of the hypotenuse in a \(30-60-90\) triangle are:
- A. \(\sqrt{3} : 2\)
- C. [tex]\(2\sqrt{3} : 4\)[/tex]
- The shortest side (opposite the \(30^\circ\) angle) is \(x\).
- The longer leg (opposite the \(60^\circ\) angle) is \(x\sqrt{3}\).
- The hypotenuse (opposite the \(90^\circ\) angle) is \(2x\).
In this problem, we are asked to find the ratio of the length of the longer leg to the length of the hypotenuse. The ratio for a \(30-60-90\) triangle is known to be:
[tex]\[ \text{Longer Leg : Hypotenuse} = x\sqrt{3} : 2x = \sqrt{3} : 2 \][/tex]
We need to verify which of the given options match this ratio.
### Option A: \(\sqrt{3} : 2\)
This matches our ratio exactly.
### Option B: \(\sqrt{3} : \sqrt{3}\)
Here, both parts of the ratio are equal, so it simplifies to \(1 : 1\), which does not match \(\sqrt{3} : 2\).
### Option C: \(2\sqrt{3} : 4\)
Let's simplify this ratio:
[tex]\[ \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \][/tex]
This matches \(\sqrt{3} : 2\).
### Option D: \(2 : 2\sqrt{2}\)
Let's simplify this ratio:
[tex]\[ \frac{2}{2\sqrt{2}} = \frac{2}{2\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \][/tex]
This does not match \(\sqrt{3} : 2\).
### Option E: \(\sqrt{2} : \sqrt{3}\)
This ratio cannot be simplified to match \(\sqrt{3} : 2\).
### Option F: \(1 : \sqrt{3}\)
This ratio can be simplified as follows:
[tex]\[ \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \][/tex]
This does not match \(\sqrt{3} : 2\).
Based on these evaluations, the ratios that match the ratio of the length of the longer leg to the length of the hypotenuse in a \(30-60-90\) triangle are:
- A. \(\sqrt{3} : 2\)
- C. [tex]\(2\sqrt{3} : 4\)[/tex]
