Mathematics — Standard (041) — CBSE Class 10 board question

Since ARC and AQB are tangents from point A to the smaller circle, and BC is a chord of the larger circle touching the smaller circle at R, by the property of concentric circles, R is the midpoint of AC (perpendicular from centre bisects the chord).

Also, by Theorem 10.2, the lengths of tangents from an external point are equal:
$$AQ = AR = 5 \text{ cm}$$

Since R is the midpoint of AC:
$$AC = 2 \times AR = 2 \times 5 = \boxed{10 \text{ cm}}$$

Source: Chapter 10, Section 10.3 (Theorem 10.2) and Example 1

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• Key property 1 (Example 1): In two concentric circles, a chord of the larger circle that is tangent to the smaller circle is bisected at the point of contact. So AR = RC.
• Key property 2 (Theorem 10.2): Tangents from an external point to a circle are equal in length. Here A is external to the smaller circle, so AQ = AR = 5 cm.
• Combining: AC = 2 × AR = 10 cm. Examiners expect both properties to be stated clearly.