Identify the sides of the isosceles triangle.

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 $x \neq 0$):

$$9 = 50 - 50 \cos \theta$$ $$50 \cos \theta = 50 - 9$$ $$50 \cos \theta = 41$$ $$$$