Struggling with the Missing Side Length with One Length

Question

In class, we have been learning about using proportions to solve for missing lengths of triangles, however, this problem has been leaving me confused. Both of the hypotenuses of the two right triangles add up to get 35. I need to find the missing length for the triangle on the left's hypotenuse. I tried YouTube and all sorts of methods, but not one that fits for this question.

Answer 1

If $x$ is the length of the hypothenuse you need to find, then by similarity, the other hypothenuse has length $8x/12$. You'll then get an equation you can easily solve.

\begin{align*} \frac{2x}{3}+x&=35\\ \frac{5x}{3}&=35\\ x&=21 \end{align*}

Answer 2

Just push up the lower triangle up by $12$ units in a simple construction to remove annoying hypotenuse intersection.

$$ \sin\theta=\dfrac{12+8}{35}=\dfrac{12}{?}\to ?=21 $$