Question

This is a 30-60-90 triangle.What is the measure of x?Rationalize the denominator.18x = [?][ ]X

Answer 1

We have a triangle with angles of measures 30°, 60° and 90°. We can place the measures in the triangle as:

We can relate angles and sides with the trigonometric ratios.

We can write:

`\begin{gathered} \cos (60\degree)=\frac{\text{Adyacent}}{\text{Hypotenuse}}=(x)/(h) \\ \sin (60\degree)=\frac{\text{Opposite}}{\text{Hypotenuse}}=(y)/(h) \\ \tan (60\degree)=\frac{\text{Opposite}}{\text{Adyacent}}=(y)/(x) \end{gathered}`

From the last ratio, we can calculate x as:

`\begin{gathered} \tan (60\degree)=(y)/(x) \\ x=(y)/(\tan(60\degree))=\frac{18}{\sqrt[]{3}}=\frac{18}{\sqrt[]{3}}\cdot\frac{\sqrt[]{3}}{\sqrt[]{3}}=\frac{18\cdot\sqrt[]{3}}{3}=6\sqrt[]{3} \end{gathered}`

Answer: x = 6√3