Question

if the longer leg in the following 30° - 60° - 90° triangle has length 8n units, what are the lengths of the other leg and the hypotenuse? note: enter the exact, fully simplified and rationalized answers.

Explanation:

Step1: Identify the longer - leg relationship

In a 30 - 60 - 90 triangle, the longer leg (opposite the 60° angle) is $\sqrt{3}x$. Given that the longer leg is $8n$, so $\sqrt{3}x = 8n$.

Step2: Solve for $x$ (the shorter leg)

We can solve the equation $\sqrt{3}x = 8n$ for $x$. Divide both sides by $\sqrt{3}$: $x=\frac{8n}{\sqrt{3}}$. Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{3}$, we get $x = \frac{8n\sqrt{3}}{3}$.

Step3: Find the hypotenuse

The hypotenuse of a 30 - 60 - 90 triangle is $2x$. Substitute $x=\frac{8n\sqrt{3}}{3}$ into $2x$, we have $2x=\frac{16n\sqrt{3}}{3}$.

Answer:

The length of the shorter leg is $\frac{8n\sqrt{3}}{3}$ units and the length of the hypotenuse is $\frac{16n\sqrt{3}}{3}$ units.