Question

what is the length of ef? 3.8 2.4 1.2 0.5

Explanation:

Step1: Find angle at E

In triangle \( DEF \), sum of angles is \( 180^\circ \). So \( \angle E = 180^\circ - 75^\circ - 50^\circ = 55^\circ \)? Wait, no, wait—wait, maybe I mixed up. Wait, the Law of Sines: \( \frac{EF}{\sin D} = \frac{DE}{\sin F} \). Let's correct: \( \angle D = 75^\circ \), \( \angle F = 50^\circ \), so \( \angle E = 180 - 75 - 50 = 55^\circ \)? Wait, no, wait, DE is length 3, opposite angle F (50°), and EF is opposite angle D (75°)? Wait, no: side DE is between D and E, length 3. So side opposite \( \angle F \) is DE (length 3), side opposite \( \angle D \) is EF. So Law of Sines: \( \frac{EF}{\sin D} = \frac{DE}{\sin F} \). So \( EF = \frac{DE \cdot \sin D}{\sin F} \).

Step2: Plug in values

\( DE = 3 \), \( \sin D = \sin 75^\circ \approx 0.9659 \), \( \sin F = \sin 50^\circ \approx 0.7660 \). So \( EF = \frac{3 \cdot 0.9659}{0.7660} \approx \frac{2.8977}{0.7660} \approx 3.78 \approx 3.8 \). Wait, but let's check again. Wait, maybe I mixed up angles. Wait, angle at D is 75°, angle at F is 50°, so angle at E is 55°? Wait, no, 75 + 50 = 125, 180 - 125 = 55. So side DE is length 3, which is opposite angle F (50°). Side EF is opposite angle D (75°). So Law of Sines: \( \frac{EF}{\sin D} = \frac{DE}{\sin F} \). So \( EF = \frac{3 \cdot \sin 75^\circ}{\sin 50^\circ} \). Calculating: \( \sin 75 \approx 0.9659 \), \( \sin 50 \approx 0.7660 \). So 3 * 0.9659 = 2.8977; 2.8977 / 0.7660 ≈ 3.78, which is approximately 3.8. So the answer should be 3.8. Wait, but let's check the options: 3.8, 2.4, 1.2, 0.5. So 3.8 is an option.

Answer:

3.8