Question 149128
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. 
<pre><font size = 4 color="indigo"><b>
{{{drawing(300,300,-6,14,-10,10, circle(0,0,5.343), locate(13.3,.4,G),
 locate(6,4,12), locate(10,.6,"48°"),locate(2.2,6,A), locate(2.2,-5,B),
line(13.13563534,0,2.173,-4.881),
line(13.13563534,0,2.173,4.881) )}}} 

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

{{{drawing(300,300,-6,14,-10,10, circle(0,0,5.343), locate(13.3,.4,G),
 locate(6,4,12), locate(10,.6,"48°"),locate(2.2,6,A), locate(2.2,-5,B),
line(13.13563534,0,2.173,-4.881), line(2.173,-4.881,0,0),line(2.173,4.881,0,0),
line(13.13563534,0,2.173,4.881), locate(-1,.6,O),locate(.5,3.1,r), locate(.5,-2,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.
 
{{{drawing(300,300,-6,14,-10,10, circle(0,0,5.343), locate(13.3,.4,G),
 locate(6,4,12), locate(2.2,6,A), locate(2.2,-5,B),
line(13.13563534,0,2.173,-4.881), line(2.173,-4.881,0,0),line(2.173,4.881,0,0),
line(13.13563534,0,2.173,4.881), locate(-1,.6,O), line(13.13563534,0,0,0),
locate(9,1.2,"24°"), locate(9,-.4,"24°"), locate(5.4,1.2,P),locate(.5,3.1,r), locate(.5,-2,r)

  )}}}

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 

{{{tangent=(opposite)/(adjacent)}}}

{{{tan(24)=r/(GA)}}}

{{{tan(24)=r/12}}}

Put 1 under the {{{tan(24)}}}

{{{tan(24)/1=r/12}}}

Cross-multiply:

{{{r=12tan(24)}}}

Next we calculate OG:

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

So we use 

{{{cosine=(adjacent)/(hypotenuse)}}}

{{{cos(24)=GA/OG)}}}

{{{cos(24)=12/OG}}}

Put 1 under the {{{cos(24)}}}

{{{cos(24)/1=12/OG}}}

Cross-multiply:

{{{OG=12cos(24)}}}

So now we can find GP by

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

GP = OG - OP = {{{12cos(24)-12tan(24)=5.61981268cm}}}

Edwin</pre>