I need help with problem 33. I don’t really know how to go about solving it. Whenever I try to express the vertices as variables, everything just ends up being really messy.
For the guy to proof of the midpoint formula given in problem 35, the proof seems a bit wishy-washy that’s not rigorous. Wouldn’t it be better to use the distance formula to show that the segment addition postulate and definition of midpoint from geometry hold?
I don’t understand your second question at all: there is no proof at all in Problem $35$, so there certainly isn’t one that’s not rigorous. The problem is for you to provide proofs of those five statements. How you do so will depend on what postulates and earlier theorems you have available. I can only guess what your segment addition postulate and definition of midpoint are, but it seems likely that they are the tools that you’re expected to use.
Once you know the result in part e of Problem $35$, Problem $33$ is just a matter of using the given information to set up a couple of systems of simultaneous equations. If the vertices are $A(x_A,y_A)$, $B(x_B,y_B)$, and $C(x_C,y_C)$, the information on the midpoints tells you that $7=\frac12(x_A+x_B)$, $3=\frac12(y_A+y_B)$, $10=\frac12(x_B+x_C)$, $9=\frac12(y_B+y_C)$, $5=\frac12(x_A+x_C)$, and $5=\frac12(y_A+y_C)$. That gives you three equations involving the three $x$-coordinates:
$$\begin{align*} x_A+x_B&=14\\ x_B+x_C&=20\\ x_A+x_C&=10\,. \end{align*}$$
This system is easy to solve: subtract the last equation from the first to find that $x_B-x_C=4$, add that to the second to get $2x_B=24$ and therefore $x_B=12$, and then use the first two equations to find $x_A$ and $x_C$. Finding the $y$-coordinates is equally easy.