Multiple choice

Two circles, each of radius 4 cm, touch externally. Each of these two circles is touched externally by a third circle. If these three circles have a common tangent, then the radius of the third circle, in cm, is

  1. 2

  2. π 3

  3. 1 2

  4. 1
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D Correct answer
Explanation

For three circles of radii r1, r2, r3 touching each other and a common tangent, the relationship is 1/sqrt(r3) = 1/sqrt(r1) + 1/sqrt(r2). Here r1=4, r2=4. 1/sqrt(r3) = 1/2 + 1/2 = 1. So sqrt(r3) = 1, r3 = 1.

AI explanation

Let the radius of the third circle be r. The centers of the three circles and their common tangent point form a right trapezoid, allowing us to equate the horizontal distances. The distance between the centers of the first two circles is 4 plus 4, which is 8 cm. Using the Pythagorean theorem for the right triangles formed with the common tangent, we equate the horizontal projections: the square root of ((4 plus r) squared minus (4 minus r) squared) plus the square root of ((4 plus r) squared minus (4 minus r) squared) equals 8. This simplifies to 2 times the square root of (16r) equals 8, so the square root of (16r) is 4. Squaring both sides gives 16r equals 16, meaning the radius r is 1.

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