In Fig. 1, there are two concentric circles with centre O. If ARC and AQB are tangents to the smaller circle from the point A lying on the larger circle, find the length of AC, if AQ = 5 cm.
Generated by claude-sonnet-4-6 · 2026-06-15 10:28 · grounding rag
Model Answer
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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Explanation
- 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.