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# help with geometry

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

Jan 21, 2022

#1
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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 = $$16\sqrt{3}\over3$$

gemetry

Jan 23, 2022
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Hint:   RQ = TQ

Jan 25, 2022