Question : If two concentric circles have radii of 5 cm and 3 cm, then the length of the chord of the larger circle that touches the smaller circle is:
Option 1: 6 cm
Option 2: 7 cm
Option 3: 10 cm
Option 4: 8 cm
Correct Answer: 8 cm
Solution :
Let O be the centre of the concentric circle of radii 5 cm and 3 cm, respectively. Let PQ be a chord of the larger circle that touches the smaller circle at point M.
OM $\perp$ PQ
In $\triangle \mathrm{OMP}$,
$\mathrm{OP^2=OM^2+PM^2}$
⇒ $\mathrm{5^2=3^2+PM^2}$
⇒ $\mathrm{PM^2 = 25-9}$
⇒ $\mathrm{PM=4}\;\mathrm{cm}$
So, $\mathrm{PQ=2PM=8\;\mathrm{cm}}$
Hence, the correct answer is 8 cm.
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