Question

find c.
right triangle with angles 30°, 60°, 90°, one leg 2 mi, hypotenuse? wait no, the side labeled c is adjacent to 30°? wait the triangle has a right angle, 30°, 60°, one leg is 2 mi (opposite 30°? wait no, lets recheck: the right angle, then 30° at the left, 60° at the bottom, the vertical leg is 2 mi. so we need to find c, which is the horizontal leg? wait, in a 30-60-90 triangle, the sides are in ratio 1 : √3 : 2, where the side opposite 30° is the shortest one. wait the vertical leg is 2 mi, which angle is it opposite to? the 30° angle? wait no, the angle at the left is 30°, so the side opposite 30° would be the vertical leg? wait no, lets label the triangle: right angle at top, 30° at left (so vertex a), right angle at b, 60° at c (bottom). so side bc is 2 mi (vertical leg), side ab is c (horizontal leg), side ac is hypotenuse. so angle at a is 30°, so side opposite angle a (which is bc) is 2 mi. in a 30-60-90 triangle, the side opposite 30° is the shortest side, length x, side opposite 60° is x√3, hypotenuse is 2x. so if bc (opposite 30°) is 2 mi, then x=2, so side opposite 60° (which is ab, labeled c) would be x√3 = 2√3? wait no, wait angle at a is 30°, so side bc is opposite angle a (30°), so bc = x = 2 mi. then side ab (c) is opposite angle c (60°), so ab = x√3 = 2√3? wait but maybe i got the sides wrong. alternatively, maybe the vertical leg is adjacent to 30°? wait, lets use trigonometry. tan(30°) = opposite/adjacent = 2/c? wait no, angle at a is 30°, so tan(30°) = bc / ab = 2 / c. tan(30°) = 1/√3, so 1/√3 = 2/c → c = 2√3. alternatively, cot(30°) = c / 2, and cot(30°) = √3, so c = 2√3. so the problem is to find c in this 30-60-90 triangle, with one leg 2 mi, find the other leg (c) in simplest radical form. write your answer in simplest radical form. blank miles

Explanation:

Step1: Identify trigonometric ratio

We use the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$
Here, $\theta = 30^\circ$, opposite side $= 2$ mi, adjacent side $= c$.
So $\tan(30^\circ) = \frac{2}{c}$

Step2: Substitute $\tan(30^\circ)$ value

We know $\tan(30^\circ) = \frac{1}{\sqrt{3}}$. Substitute:
$\frac{1}{\sqrt{3}} = \frac{2}{c}$

Step3: Solve for $c$

Cross-multiply to isolate $c$:
$c = 2\sqrt{3}$

Answer:

$2\sqrt{3}$