Line segment JH is tangent to circle G at point J. What is the value of line segment KH?

Note that KG  = 5

So....we  have a right triangle  such that

HJ^2  + GJ^2   =  (GK + HK)^2

12^2  +  5^2  = ( 5 + HK)^2

144  +  5^2   = 25  + 10 HK  + HK^2

169  = 25  +10HK + HK^2

144   = 10HK + HK^2

HK^2  + 10HK  - 144   =  0

(HK - 8)  ( HK + 18)  =  0

The first  factor  set to 0  solves this

HK  - 8  = 0

HK  =  8

(We could actually see  that we have a 5-12-13  right triangle....so HK =  13 -  5  =  8)

JH = 12          JG = GK = 5

KH = sqrt(JG² + JH²) - GK         

KH = 8 units