Given triangle KLM with two sides given as 45 yd and 20 yd and a 25 degree angle opposite the 20 yd side.
We find the angle made at the opposite of the 45 yd side using the sine rule as follows.
[tex] \frac{\sin{A}}{a} = \frac{\sin{B}}{b} \\ \frac{\sin{25^o}}{20} = \frac{\sin{B}}{45} \\ \sin{B}= \frac{45\sin{25^o}}{20} = \frac{19.0178}{20} =0.9509 \\ B=\arcsin{(0.9509)}=71.97^o[/tex]
Thus, the third angle is given by 180 - 25 - 71.97 = 83.03 degrees.
We also use the sine rule to find the third side of the triangle as follows.
[tex]\frac{\sin{A}}{a} = \frac{\sin{C}}{c} \\ \frac{\sin{25^o}}{20} = \frac{\sin{83.03^o}}{c} \\ c= \frac{20\sin{83.03^o}}{\sin{25^o}} = \frac{19.8522}{\sin{25^o}} =46.97 yds[/tex]
Therefore, amount of fencing needed to surround the perimeter of the picnic area = 45 + 20 + 46.97 = 111.97 ≈ 120 yds
