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Tangent to two circles


Two circles are drawn with five inches between the centers. A line is drawn tangent to the bottom of one circle and the top of the other. The circumference of one circle is 6.28 inches. The area of the other circle is 12.56 inches squared, Angle x is formed by the tangent line and the line between the centers. Determine angle x. (assume that pi = 3.14)

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circumference(a)=6.28=2(pi)r, yields r(a)=1

area(b)=12.56=(pi)r^2, yields r(b)=2
let the intersection of the line between the centers and the tangent line be point C.
For each circle, the center, the point of tangency, and C form a right triangle. The two triangles are mathematically similar.
Call each triangle by the same letter as the circle whose center it contains. Thus triangle A:triangle 2 is 1:2.
The hypotenuse of triangle A plus the hypotenuse of triangle B equals 5. Additionally, hypotenuse B is twice hypotenuse A. Thus, hypotenuse A is 5/3
Angle is determined by inverse sine; opposite over adjacent.

sin^(-1)(a/c)=sin^(-1)(3/5)=36.8698976 degrees or 0.643501109 radians

Basically, the tangent line, the line between the centers, and the radii form a pair of 3:4:5-ratio right triangles.

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