Question

The line AB is 6 metres and is tangent to the inner of two concentric circles at point C. Radii of the two circles are integers. What is the radius of the outer circle?

• A. 5 metres
• B. 4 metres
• C. 6 metres
• D. 3 metres

Hint:

Use Pythagoras on concentric circle tangents, then apply difference of squares to solve radius relations.

Answer 1

The Correct Option is A

Let: - \( r \) be the radius of the inner circle, - \( R \) be the radius of the outer circle, - Line AB = 6 m, - Point C is the point of tangency from AB to the inner circle.
Since both circles are concentric, and AB is a tangent to the inner circle, triangle OCB is a right triangle where: - OB = R (outer radius), - OC = r (inner radius), - CB = half of AB = 3 (since AB is tangent to both sides symmetrically). Using Pythagoras in triangle OCB: \[ OB^2 = OC^2 + CB^2 R^2 = r^2 + 3^2 = r^2 + 9 R^2 - r^2 = 9 (R - r)(R + r) = 9 \] Now factor 9 as a product of integers: \[ (1,9) R - r = 1, R + r = 9 R = 5, r = 4 \] \[ (3,3) R - r = 3, R + r = 3 R = 3, r = 0 \text{ (not valid)} \] Only valid integer solution: \[ \boxed{R = 5 \text{ metres}} \]