Question

find e.
write your answer as an integer or as a decimal rounded to the nearest tenth.

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines states that for a triangle with sides \(a\), \(b\), \(c\) and the angle \(C\) opposite side \(c\), \(c^{2}=a^{2}+b^{2}-2ab\cos C\). In \(\triangle EFG\), let \(EG = 8\), \(EF=12\), \(\angle E = 102^{\circ}\), and \(GF = e\). Then \(e^{2}=8^{2}+12^{2}-2\times8\times12\times\cos(102^{\circ})\).

Step2: Calculate each term

First, \(8^{2}=64\), \(12^{2}=144\). \(\cos(102^{\circ})\approx - 0.2079\). Then \(2\times8\times12\times\cos(102^{\circ})\approx2\times8\times12\times(- 0.2079)=-39.8208\).

Step3: Compute \(e^{2}\)

\(e^{2}=64 + 144-(-39.8208)=64+144 + 39.8208=247.8208\).

Step4: Find \(e\)

Take the square - root of \(e^{2}\), \(e=\sqrt{247.8208}\approx15.7\).

Answer:

\(15.7\)