Question

question 1(multiple choice worth 1 points) (05 01 mc) in △def, what is the length of segment df? 27

Explanation:

Step1: Identify trig - ratio

In right - triangle $DEF$, we know $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\theta = 60^{\circ}$ and the side opposite to $\angle F$ is $DE = 27$, and we want to find the hypotenuse $DF$.

Step2: Apply sine formula

We know that $\sin60^{\circ}=\frac{DE}{DF}$. Since $\sin60^{\circ}=\frac{\sqrt{3}}{2}$ and $DE = 27$, we have $\frac{\sqrt{3}}{2}=\frac{27}{DF}$.

Step3: Solve for $DF$

Cross - multiply to get $DF\times\sqrt{3}=27\times2$. Then $DF=\frac{54}{\sqrt{3}}$. Rationalize the denominator: $DF=\frac{54\sqrt{3}}{3}=18\sqrt{3}$.

Answer:

$18\sqrt{3}$