In the diagram below, $\angle PQR = \angle PRQ = \angle STR = \angle TSR$, $RQ = 8$, and $SQ = 3$. Find $PQ$. [asy] pair A,B,C,D,E; A = (0, 0.9); B = (-0.4, 0); C = (0.4, 0); D = (-0.275, 0.16); E = (0.11, 0.65); draw(A--B); draw(A--C); draw(B--C); draw(B--E); draw(C--D); label("$P$",A,N); label("$Q$", B, S); label("$R$", C, S); label("$S$", D, S); label("$T$", E, W); [/asy]
I know that triangles PQR and TSR are similar. I also THINK that QTR is similar to them as well. I can set up a similarity statement of QT/TR is similar to TR/TS which is similar to PQ/QR, but other than that, I'm stuck.
Any help is appreciated!
By similar triangles, PQ = 20*sqrt(2)/3.
Could you tell me how you got that answer?