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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!

 Mar 5, 2023
 #1
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By similar triangles, PQ = 20*sqrt(2)/3.

 Mar 5, 2023
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Could you tell me how you got that answer?

Blizzardshine  Mar 5, 2023

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