Hard Geometry Problems

2. 

minor arc TA = 2 *angle QTA =   2(47) =  94° 

minor arc UA =  2* angle  RUA  =   2 (65) = 130°      

arc TAU  =  94 + 130 =   224°

minor arc TU  = 360 - 224  = 136°

angle QXR  =  (1/2)  [ arc TAU  - arc TU ]  = (1/2) [ 224 - 136]  = 88° 

3.  Call the center of the circle, O....Let M be the mid-point of chord AB            

AO =  r

PAO  is a right triangle  with PAO  =  90°

PO  = sqrt [ AO^2  + PA^2]  = sqrt [ 2^2 + 4^2 ] = sqrt [ 20] =  2sqrt 5

Triangles

AOP  and  MOA  are similar  right triangles

OA / AP  = OM / MA

2/4  = OM / MA

1/2 = OM/ MA OM = 2MA

AO^2   =  OM^2  + MA^2

2^2  = (2MA)^2  + MA^2

4 = 5 MA^2

MA  =  2/sqrt 5

2MA =  AB  =  4 / sqrt 5  =   (4/5)sqrt 5

Both of these are inccorect,