Circle and angle

Here's a diagram : 

c)  If we draw a radius bisecting chord  PR.....it will meet PR at right angles......let E be the intersection of this radial line and  the chord

Triangle AEP is right  with  AP = 15, PE = 10

So.....AE is the distance from A  to PR.......and by the Pythagorean Theorem

AE =  √[ AP^2 - AE^2] = √ [ 15^2 - 10^2 ]  = √  [ 225 - 100]  = √125  =  5√5 units

d) Angle APR = angle APE 

And we can find the measure of angle APE as

cos APE = PE / AP =  10 / 15  =  2 /3

So

arccos (2/3)  = 48.189°

And angle APR = angle QPR......but angle RAQ  =  twice the measure of angle APE = 2 * 48.189 ≈ 96.4°  

But...in triangle RAQ, AR = AQ.....so angle ARQ = angle AQR

Angle ARQ can be found as   [ 180 - angle RAQ] / 2 =  [ 180 - 96.4 ] / 2 ≈  41.8°

YA YEEET