Question

What is the length of the hypotenuse of a 30-60-90 triangle if the side opposite the 60-degree angle is 63–√
?(1 point)

Answer 1

In a 30-60-90 triangle, the relationships between the lengths of the sides are as follows:

- The side opposite the 30-degree angle (the shortest side) is \( x \).
- The side opposite the 60-degree angle is \( x\sqrt{3} \).
- The hypotenuse (the longest side) is \( 2x \).

According to the problem, the side opposite the 60-degree angle is given as \( 63 - \sqrt{3} \). We can set up the equation:

\[
x\sqrt{3} = 63 - \sqrt{3}
\]

To find \( x \), we can isolate \( x \):

\[
x = \frac{63 - \sqrt{3}}{\sqrt{3}}
\]

Next, we need to find the hypotenuse, which is given by:

\[
\text{Hypotenuse} = 2x = 2 \cdot \frac{63 - \sqrt{3}}{\sqrt{3}}
\]

Simplifying further:

\[
\text{Hypotenuse} = \frac{2(63 - \sqrt{3})}{\sqrt{3}} = \frac{126 - 2\sqrt{3}}{\sqrt{3}}
\]

This can also be expressed as:

\[
\text{Hypotenuse} = \frac{126}{\sqrt{3}} - 2
\]

To rationalize the hypotenuse:

\[
\text{Hypotenuse} = \frac{126\sqrt{3}}{3} - 2 = 42\sqrt{3} - 2
\]

Thus, the length of the hypotenuse of the triangle is \( 42\sqrt{3} - 2 \).