The distance between the centers of two circles of radii 3 and 1 is 5. Find the length of the segment with the common inner tangent of these circles.
Let us construct the radii OA and O1B to the points of tangency A and B.
Draw a straight line O1C through point O1 until it intersects with the continuation of the radius OA.
The formed rectangle ABO1C is a rectangle, then AC = O1B = 1 cm, AC = OA + AC = 3 + 1 = 4 cm.
The length of OO1, by condition, is 5 cm.
In a right-angled triangle OO1C, according to the Pythagorean theorem, O1C ^ 2 = OO1 ^ 2 – OC ^ 2 = 25 – 16 = 9.
О1С = AB = 3 cm.
Answer: The length of the inner tangent is 3 cm.
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