To determine the length of the short leg in a [tex]\(30^{\circ}-60^{\circ}-90^{\circ}\)[/tex] triangle when the hypotenuse is known, we can utilize the properties of this special type of triangle.
In a [tex]\(30^{\circ}-60^{\circ}-90^{\circ}\)[/tex] triangle:
1. The hypotenuse is twice the length of the short leg.
2. The longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the short leg.
Here, we are given that the rafter, which forms the hypotenuse of the triangle, measures 9 feet. To find the short leg's length, we can use the property that the hypotenuse is twice the short leg.
Let’s denote the short leg by [tex]\( x \)[/tex].
According to the properties:
[tex]\[ \text{Hypotenuse} = 2 \times \text{Short Leg} \][/tex]
Given the hypotenuse is 9 feet, we can set up the equation:
[tex]\[ 9 = 2 \times x \][/tex]
To isolate [tex]\( x \)[/tex], we divide both sides of the equation by 2:
[tex]\[ x = \frac{9}{2} \][/tex]
[tex]\[ x = 4.5 \][/tex]
Thus, the length of the short leg measures [tex]\( 4.5 \)[/tex] feet.
