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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