Question

find the length of side x in simplest radical form with a rational denominator.

Answer 1

Answer:

$3\sqrt{3}$

Step-by-step explanation:

In all 30-60-90 triangles, the side lengths are in the ratio $x:x\sqrt{3}:2x$, where $x$ is the side opposite to the 30 degree angle and $2x$ is the hypotenuse of the triangle. Since the side opposite to the 30 degree angle is marked as 3, the value of $x$ must be $\boxed{3\sqrt{3}}$.

Alternatively, we can use basic trig. for a right triangle to solve. In any right triangle, the tangent of an angle is equal to its opposite side divided by its adjacent side. Thus, we have:

$\tan 60^{\circ}=\frac{x}{3},\\x=3\tan 60^{\circ}=\boxed{3\sqrt{3}}$

Answer 2

Answer:

Step-by-step explanation:

The tangent of a 60 degree angle = Sin(60) / Cos(60)

Sin(60) = Square root (3) / 2

Cos(60) = 1/2

Tan(60) = √3 / 2 // 1/2   Invert the denominator and multiply

Tan(60) = √3/2 * 2/1

Tan(60) = √3

Tan(theta) = opposite / adjacent

Tan(theta) = x/3 in this case

Tan(60) = √3 = x/3

x = 3*√3