Question 1134517
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Interesting problem.  I solved it using trigonometry; the result suggested a solution using only geometry should be possible, so I went back and found one.<br>
Let O be the center of the circumscribed circle, so OA=OB=OC=24.<br>
We are told that angle CAB is 30 degrees.  It is an angle inscribed in the circle.  Angle COB is an central angle cutting off the same arc as angle CAB, so angle COB is 60 degrees.<br>
Triangle COB is isosceles; M is the midpoint of the base.  So triangles COM and BOM are congruent right triangles, with angles COM and BOM each 30 degrees (half of angle COB).<br>
Now you have enough information to find the length of BM and CM.<br>
Finally, angle BMP is a right angle, so you can find the length of PB using the Pythagorean Theorem.