Question : The radii of two concentric circles are 34 cm and 50 cm. A and D are the points on a larger circle and B and C are the points on a smaller circle. If ABCD is a straight line and BC = 32 cm, then what is the length of AD?
Option 1: 60 cm
Option 2: 80 cm
Option 3: 75 cm
Option 4: 40 cm
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Correct Answer: 80 cm
Solution : Given: Radius $OA = 50$ cm $OB = 34$ cm $BE = \frac{BC}{2} = \frac{32}{2} = 16$ Now, in $\triangle OBE$ $OE^2=OB^2-BE^2$ $⇒OE^2=34^2-16^2$ $⇒OE^2=1156-256=900$ $\therefore OE=30$ In $\triangle OAE$ $AE^2=OA^2-OE^2$ $⇒AE^2=50^2-30^2$ $⇒AE^2=2500-900=1600$ $\therefore AE=40$ $\therefore AD = AE×2 = 40×2= 80$ Hence, the correct answer is 80 cm.
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