Circles
Positions of 2 circles
Internally Tangent
If 2 circles have only one common point which is the point of tangency then they are called internally tangent.
Given: (C) and (C') are internally tangent circles with the centers O and O' and radii R and R'
So,OO' = R-R'
Externally Tangent
If 2 circles have one point in common which is the point of tangency then they are called externally tangent.
Given:(C) and (C') are tangent externally with the centers O andO' and radii R and R'.
So OO' = R+R'
Interior Circles
If 2 circles have no points in common and one is inside the other then they are called interior circles.
Given: (C) and (C') are interior circles of centers O and O' and radii R and R' .
So OO' < R-R'
For Example
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
- 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'
Exterior Circles
Secant Circles
Iff 2 circles have 2 common points then they are called secant circles.
Given:(C) and (C') are secant circles of centers O and O' and radii R and R' .
So R-R' < OO' < R+R'