In Fig. 1, there are two concentric circles with centre O. If $A RC$ and $A QB$ are tangents to the smaller circle from the point $A$ lying on the larger circle, find the length of $A C$ , if $A Q = 5$ cm.

$Two concentric circles with centre O, tangents ARC and AQB from point A$

Question

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.

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Accepted Answer

Step 1: $AQ = AR$ (tangents drawn from external point to the circle)
$	herefore AR = 5$ cm
Step 2: Join $OR$
$	herefore OR \perp AC$ (radius tangent)
Now $AC$ is the chord of larger circle and we know that perpendicular from the centre bisects the chord
Step 3: $	herefore AR = RC = 5$ cm
$\Rightarrow AC = 5 + 5 = 10$ cm

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