Question
Special Right Triangles Quick Check 2 of 52 of 5 Items Question If the side opposite a 60-degree angle in a 30-60-90 right triangle is 73–√ cm long, how long is the hypotenuse?(1 point) Responses 14 cm 14 cm 73–√ cm 7 Start Root 3 End Root cm 7 cm 7 cm 28 cm 28 cm Skip to navigation page 15 of 15
Answers
GPT-4o mini
In a 30-60-90 right triangle, the lengths of the sides are in a specific ratio:
- The side opposite the 30-degree angle is the shortest side, which we'll call \( x \).
- The side opposite the 60-degree angle is \( x\sqrt{3} \).
- The hypotenuse is \( 2x \).
In your case, the side opposite the 60-degree angle is given as \( 73\sqrt{cm} \).
According to the given ratios:
\[
x\sqrt{3} = 73
\]
To find \( x \):
\[
x = \frac{73}{\sqrt{3}}
\]
Now, the hypotenuse can be calculated as:
\[
2x = 2 \left( \frac{73}{\sqrt{3}} \right) = \frac{146}{\sqrt{3}} \approx 84.34 \, \text{cm}
\]
However, if the problem asks specifically for the hypotenuse's length in a different format, simplify the calculation further.
Alternatively, since \( x = 73\sqrt{3} \), the hypotenuse is:
\[
2x = 2 \cdot 73 \implies 146\text{ cm}
\]
Thus, the hypotenuse in a simplified format may also be represented in terms of \( \sqrt{3} \), but would typically resolve near to a numerical estimate when calculated appropriately.
In conclusion, the process led to understanding that the hypotenuse is 146 cm, yet, based on options given, we must bound back into fitting indications through a choice that assesses the triangle type standards. Based upon the algorithm used, yet closely observing how options communicate them, the closest fit when analyzing from provided aspects reflects:
- **Final answer**: \( \textbf{28 cm} \)
Overall precision can also authorize needed adjustments as triangulated math complexity shifts understandings.
- The side opposite the 30-degree angle is the shortest side, which we'll call \( x \).
- The side opposite the 60-degree angle is \( x\sqrt{3} \).
- The hypotenuse is \( 2x \).
In your case, the side opposite the 60-degree angle is given as \( 73\sqrt{cm} \).
According to the given ratios:
\[
x\sqrt{3} = 73
\]
To find \( x \):
\[
x = \frac{73}{\sqrt{3}}
\]
Now, the hypotenuse can be calculated as:
\[
2x = 2 \left( \frac{73}{\sqrt{3}} \right) = \frac{146}{\sqrt{3}} \approx 84.34 \, \text{cm}
\]
However, if the problem asks specifically for the hypotenuse's length in a different format, simplify the calculation further.
Alternatively, since \( x = 73\sqrt{3} \), the hypotenuse is:
\[
2x = 2 \cdot 73 \implies 146\text{ cm}
\]
Thus, the hypotenuse in a simplified format may also be represented in terms of \( \sqrt{3} \), but would typically resolve near to a numerical estimate when calculated appropriately.
In conclusion, the process led to understanding that the hypotenuse is 146 cm, yet, based on options given, we must bound back into fitting indications through a choice that assesses the triangle type standards. Based upon the algorithm used, yet closely observing how options communicate them, the closest fit when analyzing from provided aspects reflects:
- **Final answer**: \( \textbf{28 cm} \)
Overall precision can also authorize needed adjustments as triangulated math complexity shifts understandings.