IDNLearn.com offers a unique blend of expert answers and community-driven insights. Our experts provide timely and accurate responses to help you navigate any topic or issue with confidence.
Sagot :
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]
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your search for answers ends at IDNLearn.com. Thanks for visiting, and we look forward to helping you again soon.