SOLUTION: An isosceles triangle with base 6 cm and base angles 30 degrees each is inscribed in a circle. A second circle,which is situated outside the triangle, touches the first circle and

Question 389514: An isosceles triangle with base 6 cm and base angles 30 degrees each is inscribed in a circle. A second circle,which is situated outside the triangle, touches the first circle and also touches the base of the triangle at its midpoint. Find its radius.
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
If you construct a diagram of the problem, you will see that the distance from the center of the large circle to the apex of the isosceles triangle is equal to the radius,R, of the large circle.
Similarly, the end points of the triangle are also equal to the radius of the large circle.
This forms an isosceles triangle but it becomes an equalateral triangle because the third side is also equal to the two other sides. The angles given prove this. I wish I could have drawn this diagram here, but I do not know how to do it with this format.
Given this analysis, the diameter,D, of the small circle is equal to the radius R of the large circle plus the distance,y, from the midpoint of the base of the given isosceles triangle to the center of the large circle.
D=y+R
tan 30 =y/3
y=3*tan (30)=1.73
Cos 30 = 3/R
R = 3/ cos 30=3.46
D=y+R=1.73+3.46=5.19
ans: radius of small circle = D/2 =2.6 cm
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