Geometry

Both questions begin by making use of the Pythagoras theorem.

For the first one, start by saying that 9^2 + r^2 = (6 + r)^2, tidy up and solve for r.

The second one is asking you if the triangle is rightangled (Pythagoras again), if it is then ST will be a tangent.

Q1:

(6+r)² = 9² + r² 

6² + 2*6*r+r² =  9² + r²

6² + 2*6*r =  9² 

2*6*r =  9² - 6²

r = ( 9² - 6² ) / ( 2*6 )

Q2: 

by using pythagoras rule

if    RT² = RS² + ST²    is true , ST is tangent to the circle.

         (5+8)²   =    5² + 12²

           (13)²   =    5² + 12²

            169    =    25 + 144

            169    =    169

yas