$(8$ marks$)$ You are the lead designer for a company that makes $4$ linkage mechanisms. All of the $r_1$ linkages that you manufacture have a length of ${15}^{\text{'}\text{'}}.$ You have a bunch of leftover $r_3$ linkages that all have a length of $20".$

You task is to design a $4-$bar linkage mechanism so that the angle $=75,$ when ${}_2=45$ and $r_2={10}^{\text{'}\text{'}}.$ Determine the lengths of $z_{,}{r}_4$ and the angles $,,$ and ${}_4.$

B