Circles

Positions of 2 circles

A circle is the locus of non collinear points equidistant from the center of the circle. Consider a circle C with center O , consider another circle C' with center O'. The line OO' is called the line of centers. When there are 2 given circles they have a certain position which can be exterior, interior, externally tangent, internally tangent, or secant.

For Example

(C) and (C') have 2 points in common A and B , and OO' is the perpendicular bisector of AB.

1) What is the position of the 2 circles (C) and (C')?

2) Deduce the measure of OO'.


Answers:


1) Since (C) and (C') intersect at 2 points A and B and OO' is the perpendicular bisctor of AB (given) then (C) and (C') are secant

2) Since (C) and (C') are secant then R-R'<OO'<R+R'

The converse

The converse states that:

  • If OO' > R + R', then C and AC' are exterior circles
  • If OO' < R - R', then C and C' are interior circles with R>R'
  • If OO' = R + R', then C and C' are externally tangent circles
  • If OO' = R - R', then C and C' are internally tangent circles with R>R'
  • If R - R' < OO' < R + R', then C and C' are secant circles with R > R'

Here are some real life examples
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I hope you enjoyed my flyer and learned how the position of 2 circles is useful and helpful.