In the diagram below, $\angle PQR = \angle PRQ = \angle STR = \angle TSR$, $RQ = 8$, and $SQ = 2$. 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]

Please Help Fast!!!!

And Yes, I know about the other question. Need help fast.

Thanks in advance!

Guest Oct 31, 2020

#1**+1 **

We can form isosceles triangles with our angles.

We know that RQ is 8, and SQ is 2.

However, we can't solve this problem without finding another isosceles triangle! This calls for a diagram!

We can see now that ∆QTR is isosceles, with QT = QR = 8.

But now we know that QT = 8, so ST = 8 - QS = 8 - 2 = 6!

By AA similarity, ∆QTR is similar to ∆RST! By similar triangle ratios, RT/8 = 6/RT, and RT^2 = 48.

Thus, RT = 4 √3.

Also, by the base angles being equal, by AA similarity we have ∆PQR is similar to ∆QRT, and thus we have PQ/8 = 8/4√3 , so PQ = 64/4√3.

Rationalizing the denominators, we have PQ = $\frac{16√3 }{3}$

Pangolin14 Oct 31, 2020