12. (a) Step 1: Identify the sides of the isosceles triangle. The ratio of the sides is $5:3:5$. This means the two equal sides have a length proportional to 5, and the third side has a length proportional to 3. Let the sides be $a=5x$, $b=5x$, and $c=3x$. We need to find the angle $\theta$ between the two equal sides (i.e., the angle opposite side $c$). Step 2: Apply the Law of Cosines. The Law of Cosines states $c^2 = a^2 + b^2 - 2ab \cos \theta$. Substitute the side lengths: $$(3x)^2 = (5x)^2 + (5x)^2 - 2(5x)(5x) \cos \theta$$ $$9x^2 = 25x^2 + 25x^2 - 50x^2 \cos \theta$$ $$9x^2 = 50x^2 - 50x^2 \cos \theta$$ Step 3: Solve for $\cos \theta$. Divide both sides by $x^2$ (assuming $x \neq 0$): $$9 = 50 - 50 \cos \theta$$ $$50 \cos \theta = 50 - 9$$ $$50 \cos \theta = 41$$ $$\cos