In a [tex]\(30^\circ\)[/tex]-[tex]\(60^\circ\)[/tex]-[tex]\(90^\circ\)[/tex] triangle, the sides are in a specific ratio based on the angles of the triangle. Let's go through this step-by-step:
1. Identify the side ratios:
In a [tex]\(30^\circ\)[/tex]-[tex]\(60^\circ\)[/tex]-[tex]\(90^\circ\)[/tex] triangle, the sides opposite these angles have a unique ratio. The ratio of the sides opposite the [tex]\(30^\circ\)[/tex], [tex]\(60^\circ\)[/tex], and [tex]\(90^\circ\)[/tex] angles is [tex]\(1 : \sqrt{3} : 2\)[/tex].
2. Assign the shorter leg:
Let's denote the length of the shorter leg (opposite the [tex]\(30^\circ\)[/tex] angle) by [tex]\(x\)[/tex].
3. Find the longer leg:
The longer leg in this triangle is opposite the [tex]\(60^\circ\)[/tex] angle. According to the ratio, the longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the shortest leg.
Thus, the longer leg = [tex]\(x \cdot \sqrt{3}\)[/tex].
To express [tex]\(\sqrt{3}\)[/tex] in a numerical format (which is approximately [tex]\(1.7320508075688772\)[/tex]):
[tex]\[ \text{Longer leg} = x \cdot 1.7320508075688772 \][/tex]
Therefore, if the shorter leg in a [tex]\(30^\circ\)[/tex]-[tex]\(60^\circ\)[/tex]-[tex]\(90^\circ\)[/tex] triangle is denoted by [tex]\(x\)[/tex], then the longer leg is [tex]\(x \cdot \sqrt{3} \approx x \cdot 1.7320508075688772\)[/tex].
