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

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) (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:

Start with the cosine function.

Plug in the given sides and angle.

Multiply both sides by

Divide both sides by

Evaluate to get

Multiply by the reciprocal

Cancel like terms.

Simplify

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

Start with the sine function.

Plug in the angle 30 , the opposite side AO, and the hypotenuse 24

Take the sine of 30 to get

Cross multiply.

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

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