geometry

BA is tangent to the circle at A and angle DBA = 90 degrees. If AB = 15 and CD = 10, find the radius of the circle.

r = 1/2 [sqrt(10² + 30²)]             ( 10 and 30 are the lengths of the legs of an inscribed right-angled triangle )

We can use  the tangent-secant theorem to find   CB

We  have  that

AB^2  = CB  *( CB + DC)

15^2  =  CB ( CB  + 10)

225 = CB^2  + 10CB

CB^2  + 10CB  -225  =  0

Solving this quadratc (taking the positive result) gives us  that CB  =   5sqrt (10)  - 5

So  the  radius =  (1/2)DC  + CB   =   (1/2) 10  +  5sqrt (10)  - 5  =

5 + 5sqrt (10)   - 5   = 

5sqrt (10)