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# Q7. Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Let O be the common centre of two concentric circles, and let AB be a chord of the larger circle touching the smaller circle at P. Join OP.

∵ OP is the radius of the smaller circle and AB is tangent to this circle at P,

∴ $$OP \ ⊥ \ AB$$.
We know that, the perpendicular drawn from the centre of a circle to any chord of the circle bisects the chord.

So, $$OP \ ⊥ \ AB$$ and $$AP \ = \ BP$$.

In $$∆ \ APO$$, we have

$$OA^2 \ = \ AP^2 \ + \ OP^2 \ => \ 5^2 \ = \ AP^2 \ + \ 3^2$$

$$AP^2 \ = \ 5^2 \ - \ 3^2 \ = \ 25 \ - \ 9 \ = \ 16$$
$$AP \ = \ 4$$

Now, $$AB \ = \ 2AP$$      [∵ $$AP \ = \ BP$$ ]

$$AB = \ 2 \ × \ 4 \ = \ 8$$

∴ The length of the chord of the larger circle which touches the smaller circle is 8 cm.