The radii of two concentric circles with centre 'O' is 12 cm and 18 cm, respectively. AB is a chord on the larger circle that is tangent to the smaller circle making contact at point 'C'. If AD is a diameter of the larger circle, then find the length of CD.
Show answer & explanation
Correct answer: B
Explanation
Setup:
- Smaller circle radius (OC) = 12 cm, larger circle radius (OA) = 18 cm
- AB is tangent to smaller circle at C, so OC ⊥ AB, and C is the midpoint of AB
Step 1: Find AC (half of chord AB)
In right triangle OCA:
AC = √(OA² − OC²) = √(18² − 12²) = √(324 − 144) = √180 = 6√5 cm
So AC = BC = 6√5 cm
Step 2: Use the diameter property
Since AD is a diameter of the larger circle and B lies on the circle, angle ABD = 90° (angle in a semicircle).
Find AB and AD:
- AB = 2 × AC = 12√5 cm
- AD = 2 × 18 = 36 cm
In right triangle ABD:
BD = √(AD² − AB²) = √(36² − (12√5)²) = √(1296 − 720) = √576 = 24 cm
Step 3: Find CD
Since C lies on line AB (same line as B), angle CBD is also 90° (same right angle as angle ABD).
In right triangle CBD:
CD = √(BC² + BD²) = √((6√5)² + 24²) = √(180 + 576) = √756