SOLUTION: the segments GA and BG are tangent to a circle at A and B, and AGB is a 48 degree angle. Given that GA = 12 cm, find the distance from G to the nearest point on the circle.

 

Label the center of the circle O. and 
draw radii to A and B. Let the radius be r



Next draw OG which bisects the 48° angle G into
two 24° angles. Let P be the point where OG intersects
the circle.  P is the nearest point on the circle to G, 
so GP is the distance we're looking for.

Plan: Calculate the radius OA and the hypotenuse OG using the 
upper right triangle using trig ratios. Then calculate OG. Then 
since OP is also a radius, we will subtract the radius OP from 
OG and get GP.
 


In the right triangle AOG, radius AO is the side opposite
angle AGO which is 24°.  GA is the side adjacent to angle AGO.

So we use 







Put 1 under the 



Cross-multiply:



Next we calculate OG:

---
OG is the hypotenuse, GA is the opposite side of 24°

So we use 







Put 1 under the 



Cross-multiply:



So now we can find GP by

subtraction, since OP = r = 12tan(24)

GP = OG - OP = 

Edwin