To solve the problem, we need to analyze the properties of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle. The sides of a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle have a specific ratio which is always [tex]\(1 : \sqrt{3} : 2\)[/tex]. Here, the sides are as follows:
1. The side opposite the [tex]\(30^\circ\)[/tex] angle (the short leg) is the smallest and is denoted as [tex]\(x\)[/tex].
2. The side opposite the [tex]\(60^\circ\)[/tex] angle (the long leg) is [tex]\(x\sqrt{3}\)[/tex].
3. The side opposite the [tex]\(90^\circ\)[/tex] angle (the hypotenuse) is the longest and is [tex]\(2x\)[/tex].
Given that the hypotenuse (which is opposite the [tex]\(90^\circ\)[/tex] angle) measures 9 feet, we can use the ratio to find the short leg.
Since the hypotenuse is [tex]\(2x\)[/tex] and it is given to be 9 feet, we can set up the equation:
[tex]\[ 2x = 9 \][/tex]
To find [tex]\(x\)[/tex]:
[tex]\[ x = \frac{9}{2} \][/tex]
Therefore, the measure of the short leg ([tex]\(x\)[/tex]) is:
[tex]\[ x = 4.5 \][/tex]
So, the short leg of the [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle measures 4.5 feet.
