The distance between the centers of two nonintersecting circles is 20cm. The length of a common external tangent segment is 16cm. If the radius of the smaller circle is 2cm, what is the radius of the larger circle?
The distance between the centers of two nonintersecting circles is 20cm. The length of a common external tangent segment is 16cm. If the radius of the smaller circle is 2cm, what is the radius of the larger circle?
I did not solve this problem, but I noticed one super important thing which might be valuable in this problem.
Let O be the center of first first with radius 2 cm.
Let C be the center of the second circle.
Draw OC, which we are told is 20 cm.
Draw External Tangent at A on circle O and B on circle C.
Draw AC which we know is 16 cm.
Let P be the intesection of OC and AB.
Then triangle OAP is similar to triangle PCB. *
*)Because <OAP = <PCB = 90 because it is tangent and hence perpendicular with radius. And <AP0 = <CPB because they are vertical angles.
See the diagram below.
Let A be the center of the smaller circle and B be the center of the smaller circle
Let C be the point where the common tangent cuts the circumference of the smaller circle and D be the point where the common tangent cuts the larger circle
Let the radius of the larger circle be x
I have constructed a diagram of the circles as you described them. Now if we remove the circles and consider the triangles only, we get the second diagram. we want the value of x, so we can use Pythagoras' theorem.
Now in triangle ABE:
(AB)^2 = (BD)^2 + (AE)^2
20^2 = (x + 2)^2 + 16^2
=> (x + 2)^2 + 16^2 - 20^2 = 0
=> (x + 2)^2 - 144 = 0
=> (x + 2)^2 = 144
=> x + 2 = sqrt(144) = 12
=> x = 12 - 2
so x = 10
therefore, the radius of the larger circle is 10 cm