In the diagram below, angle PQR = angle PRQ = angle STR = angle TSR, RQ = 8, and SQ = \(2\). Find PQ.
+1622 Given the angles, we can conclude that triangles TSR ~ triangle TRQ ~ triangle RQP.
Using base angles theorem, these 3 triangles are all isosceles, and QR = 8. That means QT = 8, and QS = 2 so ST = 6.
The ratio of QS / ST is also the ratio of PT / TR. Which is 1 : 3
Now after we have concluded the information above, we can set segment TR as \(x\).
Since we know there are similiar triangles, and we know TS = 6 and QR = 8, then we can say \({x\over6} = {8\over x}\).
Solving for x we get \(x = 4\sqrt{3}\). PT = one third of TR, so PT = \(4\sqrt{3}\over3\).
PQ = PT + PR.
Plugging in the values we get:
PQ = \(16\sqrt{3}\over3\)
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