Question

the nearest whole number.
iii. use the law of sines and law of cosines to

1. find the missing dimensions of the triangle

below. round your answers to the nearest
whole number.

triangle with vertices d, e, f: angle at d is 20°, angle at e is 138°, side ef is 8, side df is 19

Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^\circ\). Given \(\angle D = 20^\circ\) and \(\angle E = 138^\circ\), we find \(\angle F\) as:
\(\angle F=180^\circ - 20^\circ - 138^\circ = 22^\circ\)

Step2: Use Law of Sines to find side \(DE\)

Law of Sines: \(\frac{DE}{\sin F}=\frac{DF}{\sin E}\)
We know \(DF = 19\), \(\angle F = 22^\circ\), \(\angle E = 138^\circ\). So:
\(DE=\frac{DF\times\sin F}{\sin E}=\frac{19\times\sin22^\circ}{\sin138^\circ}\)
\(\sin22^\circ\approx0.3746\), \(\sin138^\circ=\sin(180^\circ - 42^\circ)=\sin42^\circ\approx0.6691\)
\(DE=\frac{19\times0.3746}{0.6691}\approx\frac{7.1174}{0.6691}\approx10.64\approx11\)

Step3: Use Law of Sines to find side \(EF\) (Wait, \(EF = 8\) is given? Wait, no, maybe find \(EF\)? Wait, no, \(EF\) is 8? Wait, the triangle: vertices D, E, F. \(EF = 8\)? Wait, no, in the diagram, \(EF = 8\), \(DF = 19\). Wait, maybe I misread. Wait, let's re - check. Wait, the sides: \(EF = 8\), \(DF = 19\), angle at E is \(138^\circ\), angle at D is \(20^\circ\). Wait, maybe find side \(DE\) (we did), and side \(EF\) is 8, and we can also find side \(DF\)? No, \(DF\) is 19. Wait, maybe the missing side is \(DE\) and the missing angle is \(\angle F\). Wait, we found \(\angle F = 22^\circ\), and \(DE\approx11\). Also, let's check finding \(EF\) (if we thought \(EF\) is missing, but no, \(EF = 8\) is given). Wait, maybe the triangle has sides: \(DF = 19\), \(EF = 8\), angle at E: \(138^\circ\), angle at D: \(20^\circ\), angle at F: \(22^\circ\), and side \(DE\approx11\).

Wait, maybe another approach. Wait, let's re - express the Law of Sines for side \(EF\) (if \(EF\) was unknown, but in the diagram \(EF = 8\). Wait, maybe the problem is to find \(DE\) and \(\angle F\) and maybe \(EF\) is 8 (given). Wait, the problem says "Find the missing dimensions". So angles: \(\angle F = 22^\circ\), sides: \(DE\approx11\).

Answer:

• \(\angle F = 22^\circ\)
• \(DE\approx11\)