Multiple choice

The radii of two concentric circles are 12 cm and 6 cm. AB is the diameter of the bigger circle and BD is a tangent to the smaller circle touching it at D and the bigger circle at E in such a way that it forms an angle of 90 degree at E with A. Point A is joined to D. The length of AD is -

  1. 8 cm

  2. 3√3 cm

  3. 6√7 cm

  4. 12 cm

  5. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In the circle, let O be the center. Triangle OED is a right triangle with OE=6 and OD=6 (radius). However, the geometry description is complex; using the property of tangents and the right angle at E, AD can be calculated via the Pythagorean theorem in the resulting triangles as 6*sqrt(7).

AI explanation

Since AB is the diameter of the larger circle with radius 12 cm, its total length is 24 cm, making the center of the circles the midpoint of AB. The tangent BD is perpendicular to the radius of the smaller circle at the point of tangency D, creating a right triangle where the hypotenuse is the center-to-B distance of 12 cm and the radius is 6 cm. Using the Pythagorean theorem, the distance from the center to D is the square root of (12 squared minus 6 squared), which equals 6 times the square root of 3 cm. Triangle OAD has side OA of 12 cm, side OD of 6 times the square root of 3 cm, and angle AOD of 150 degrees; using the cosine rule, AD squared equals 144 plus 108 minus 2 times 12 times 6 times the square root of 3 times cos(150 degrees), resulting in AD being 6 times the square root of 7 cm.

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