Question 626521
<pre>
Since AB=25 and AC=16, then BC = AB-AC = 25-16 = 9

{{{drawing(300,300,-13.5,13.5,-13.5,13.5, circle(0,0,12.5), line(3.5,12,3.5,-12),
line(-12.5,0,12.5,0),locate(-13.2,.5,A),locate(12.8,.5,B), locate(3.3,13.4,P),
locate(3.3,-12.4,Q),rectangle(3.5,0,4.5,1),locate(2.2,0,C),locate(-4.5,0,16),
locate(8,0,9)


 )}}}

Draw in AP and BP

{{{drawing(300,300,-13.5,13.5,-13.5,13.5, circle(0,0,12.5), line(3.5,12,3.5,-12),green(line(-12.5,0,3.5,12),line(3.5,12,12.5,0)),
line(-12.5,0,12.5,0),locate(-13.2,.5,A),locate(12.8,.5,B), locate(3.3,13.4,P),
locate(3.3,-12.4,Q),rectangle(3.5,0,4.5,1),locate(2.2,0,C),locate(-4.5,0,16),
locate(8,0,9)


 )}}}

&#8736;APB is a right angle because it is inscribed in a semicircle.

The three right triangles &#5123;APB, &#5123;ACP and &#5123;PCB are all similar 
because their corresponding angles are equal.  Therefore 

{{{AC/PC}}} = {{{PC/BC}}}

{{{16/PC}}} = {{{PC/9}}}

Cross-multiplying:

PC˛ = 144
 PC = 12

By symmetry, PC = QC, so PQ = 12ˇ2 = 24.

Edwin</pre>