In a 30-60-90 triangle, the ratio of the lengths of the sides is:
- The side opposite the 30-degree angle: \(x\)
- The side opposite the 60-degree angle: \(x\sqrt{3}\)
- The hypotenuse: \(2x\)
Given that the side opposite the 60-degree angle is \(63 - \sqrt{63}\), we can set up the equation:
\[ x\sqrt{3} = 63 - \sqrt{63} \]
To find the value of \(x\), we can divide both sides by \(\sqrt{3}\):
\[ x = \frac{63 - \sqrt{63}}{\sqrt{3}} \]
Now, we need to find the hypotenuse, which is \(2x\):
\[ \text{Hypotenuse} = 2x = 2 \cdot \frac{63 - \sqrt{63}}{\sqrt{3}} = \frac{2(63 - \sqrt{63})}{\sqrt{3}} \]
Thus, the length of the hypotenuse of the triangle is:
\[ \frac{126 - 2\sqrt{63}}{\sqrt{3}} \]
To express this in simplified terms:
\[ \text{Hypotenuse} = \frac{126}{\sqrt{3}} - \frac{2\sqrt{63}}{\sqrt{3}} \]
You can also rationalize the first term:
\[ \frac{126}{\sqrt{3}} = 42\sqrt{3} \]
So the final expression for the hypotenuse is:
\[ 42\sqrt{3} - \frac{2\sqrt{63}}{\sqrt{3}} = 42\sqrt{3} - 2\sqrt{21} \]
However, for the exact numeric approximation:
\[ \text{Hypotenuse} = \frac{126 - 2\sqrt{63}}{\sqrt{3}} \]
This represents the length of the hypotenuse in simplified radical form.
For a numerical solution, replace \(\sqrt{63}\) approximating it by \(7.94\):
Thus,
\[ \text{Hypotenuse} \approx 63.47 \]
This can be verified further with actual numeric computations. However, the length in terms of exact radicals is most relevant in geometry contexts.
1 answer