In a 30-60-90 triangle, the ratios of the lengths of the sides are as follows:
- The side opposite the 30-degree angle (shorter leg) is \( x \).
- The side opposite the 60-degree angle (longer leg) 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 side ratios, this side (longer leg) can be expressed as:
\[ x\sqrt{3} = 73 - \sqrt{cm} \]
To find \( x \), we can rearrange the equation:
\[ x = \frac{73 - \sqrt{cm}}{\sqrt{3}} \]
Now, we can calculate the hypotenuse \( 2x \):
\[ 2x = 2 \left( \frac{73 - \sqrt{cm}}{\sqrt{3}} \right) = \frac{2(73 - \sqrt{cm})}{\sqrt{3}} \]
Thus, the hypotenuse is:
\[ \frac{146 - 2\sqrt{cm}}{\sqrt{3}} \text{ cm} \]
This gives you the length of the hypotenuse in terms of \( \sqrt{cm} \). If you plug in the value of \( cm \), you can obtain a numerical result.
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