In the diagram below, $\angle PQR = \angle PRQ = \angle STR = \angle TSR$, $RQ = 8$, and $SQ = 3$. Find $PQ$. Sadly I cant upload my picture (it wont let me), but if you search this up in google this site will have a slightly different problem but the same pic.

Guest Jan 23, 2023

#3**0 **

ik im the one who asked this but i solved it and the answer is: 16*(sqrt10)/5 Since triangle $RQT$ is isosceles, we have \[ST = QT - QS = QR - QS = 8 - 3 = 5.\] From $\triangle QTR \sim\triangle RST$, we have $\frac{RT}{ST}=\frac{QR}{RT},$ so \[RT^2 = ST \cdot QR = 5 \cdot 8 = 40.\]Hence, $RT = \sqrt{40} = 2 \sqrt{10}.$ From $\triangle PQR \sim \triangle QRT$, we have $\frac{PQ}{QR}=\frac{QR}{RT}$, so that \[PQ=\frac{QR^2}{RT}=\frac{8^2}{2 \sqrt{10}}=\boxed{\frac{16\sqrt{10}}{5}}.\]

Guest Jan 25, 2023