SOLUTION: 10. The segments GA and GB are tangent to a circle with center O at A and B, and AGB is a 60-degree angle. Given that GA = 12 square root 3 cm, find the distance GO. Find the dista

Algebra ->  Formulas -> SOLUTION: 10. The segments GA and GB are tangent to a circle with center O at A and B, and AGB is a 60-degree angle. Given that GA = 12 square root 3 cm, find the distance GO. Find the dista      Log On


   

Question 149173: 10. The segments GA and GB are tangent to a circle with center O at A and B, and AGB is a 60-degree angle. Given that GA = 12 square root 3 cm, find the distance GO. Find the distance from G to the nearest point on the circle.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

First let's draw the picture (take note that I'm cutting AGB in half to make 2 30 degree angles):


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From the drawing, we can see that segments GA and AO are legs of a triangle while GO is the hypotenuse. Since the angle that we are working with is the 30 degree angle, this means that GA is the adjacent side. So to find the hypotenuse, we need to use the cosine function:


cos%28x%29=adjacent%2Fhypotenuse Start with the cosine function.


cos%2830%29=12%2Asqrt%283%29%2FGO Plug in the given sides and angle.


GO%2Acos%2830%29=12%2Asqrt%283%29 Multiply both sides by GO


GO=%2812%2Asqrt%283%29%29%2Fcos%2830%29 Divide both sides by cos%2830%29


GO=%2812%2Asqrt%283%29%29%2F%28sqrt%283%29%2F2%29 Evaluate cos%2830%29 to get sqrt%283%29%2F2


GO=%2812%2Asqrt%283%29%29%2A%282%2Fsqrt%283%29%29 Multiply by the reciprocal


GO=%2812%2Across%28sqrt%283%29%29%29%2A%282%2Fcross%28sqrt%283%29%29%29 Cancel like terms.

GO=12%2A2 Simplify


GO=24 Multiply

So the length of GO is 24 units.


The closest point to G will lie on the line GO. So to find the distance from G to this point, we need to find the radius.

Using the previous drawing, we can see that the radius is AO. To find the length of AO, note that AO is the opposite leg to the angle 30 degrees. So let's use the sine function to find AO



sin%28x%29=opposite%2Fhypotenuse Start with the sine function.


sin%2830%29=AO%2F24 Plug in the angle 30 , the opposite side AO, and the hypotenuse 24


1%2F2=AO%2F24 Take the sine of 30 to get 1%2F2


24=2AO Cross multiply.


12=AO Divide both sides by 2.

So the length of AO (the radius) is 12 units.


Now simply subtract the radius from the length GO to get 24-12=12

So the distance from G to the nearest point on the circle is 12 units.