A circle is centered at O and has an area of 48/pi Let Q and R be points on the circle, (ran out of space look at description)

Question

and let P be the circumcenter of triangle QRO. If P is contained in triangle QRO, and triangle PQR is equilateral, then find the area of triangle PQR.

i tried drawing it out and looking at the problem from so many different angles (pun intended :) but yes i do need some help!

Answer 1

I cannot give you the whole answer, but, can tell you how to find it (after much playing around with DESMOS):

If you represent O as (0,0) and Q as (0,r), then P will be at the intersection of y=r/2 with

y=x tan 75° and you will need some geometry and trig to get the number you need.

Oh, I forgot r = the radius of the circle which has area 48/π.

I have not worked out exactly why is this true, but I am sure it is correct.

I"m sorry it took so long. If when you review this in class, you get a better, simpler answer, please let me know.

Added:

The critical observation is that angle QPR and angle QOR are measured by the same arc so that angle QOR=30 °. Then a little geometry gets you angle POQ is also 15° and a little trig gets you the length of PR=r/(2 cos 15°).