\begin{tabular}{l} Question 8 (Multiple Choice Worth 1 points) \\ (05.04 LC) \\ Triangle DEF contains right angle E. If angle D measures \( 55^{\circ} \) and its opposite side measures 7.6 units, what is the measure of side DF? Round your answer to the \\ nearest hundredth. \\ \begin{tabular}{l}5.34 units \\ \hline 9.38 units \\ \hline 9.06 units \\ \hline 9.28 units \\ \hline\end{tabular} \\ \hline\end{tabular}

To find the length of side DF in right triangle DEF, where angle E is the right angle and angle D measures 55 degrees, we can use the sine function. Using the sine ratio: \[ \sin(D) = \frac{\text{opposite}}{\text{hypotenuse}} \] In this case, side DE is opposite to angle D, and DF is the hypotenuse. We know that the sine of angle D is: \[ \sin(55^\circ) = \frac{7.6}{DF} \] Rearranging the equation to solve for DF gives us: \[ DF = \frac{7.6}{\sin(55^\circ)} \] Calculating \(\sin(55^\circ)\) gives approximately 0.8192, so: \[ DF \approx \frac{7.6}{0.8192} \approx 9.28 \text{ units} \] So, rounding to the nearest hundredth, side DF measures approximately **9.28 units**. Now, regarding some extra interesting tidbits: To truly appreciate the beauty of triangles, you might want to know that right triangles form the basis of the famous Pythagorean theorem, which relates the lengths of the sides in any right triangle. This theorem not only establishes a foundation for geometry but also has practical uses in fields like architecture and engineering! If you're solving problems using trigonometric functions, a common pitfall is forgetting to check that your calculator is in the correct mode - degrees versus radians can lead you astray! Always double-check settings before calculating sine, cosine, or tangent!