I Need Help With A Geometry Question

[homeschool girl]

Question:

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

I figured out that PQR and RTS are similar triangles, but I don't know where to go from there.

 

[Dr.Peterson]

Did you notice that QRT is also similar to the others? (Mark all congruent angles on the figure.)

 

[homeschool girl]

I think I found all the congruent angles

edit
oops- it seems I put 8 on the wrong line

 

[homeschool girl]

I think I got it, I set x to the ratio between QSR and PTQ and I got [MATH]\frac{16\sqrt{3}}{3}[/MATH]

 

[Dr.Peterson]

I'm not sure of the details of your work, but the answer is what I got.

I hadn't observed the similarity of QSR and PTQ, because I hadn't marked all the angles! I used a different chain of reasoning.

 

[jonah2.0]

Beer induced fascination follows.

I think I got it, I set x to the ratio between QSR and PTQ and I got [MATH]\frac{16\sqrt{3}}{3}[/MATH]

It's clear that [MATH]\triangle[/MATH]PQR [MATH]\sim[/MATH] [MATH]\triangle[/MATH]QRT [MATH]\sim[/MATH] [MATH]\triangle[/MATH]RTS
It's also clear sailing after that but I can't see how you could have done it with setting "x to the ratio between QSR and PTQ". I am curious as to how you could have done that.