Question : The radii of two concentric circles are 13 cm and 8 cm. AB is the diameter of the bigger circle and BD is a tangent to the smaller circle touching it at D and the bigger circle at E. Point A is joined to D. The length of AD is:
Option 1: 20 cm
Option 2: 19 cm
Option 3: 18 cm
Option 4: 17 cm
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Correct Answer: 19 cm
Solution :
Let the line BD intersect the bigger circle at point C.
Join AC.
OD $\perp$ BD ⇒ OD $\perp$ BC
So, BD = DC
⇒ D is the midpoint of BC.
Also, O is the midpoint of AB.
In $\triangle$BAC, O is the midpoint of AB and D is the midpoint of BC.
We know that segments joining the midpoints of any two sides of a triangle are half of the third side.
So, OD = $\frac{1}{2}$AC ⇒ AC = 2OD ⇒ AC = 2 × 8 = 16 cm
In $\triangle$OBD, OB
2
= BD
2
+ OD
2
⇒ BD
2
= $13^2-8^2$
⇒ BD = $\sqrt{105}$ = DC
In $\triangle$ADC, AD
2
= DC
2
+ AC
2
⇒ AD
2
= $(\sqrt{105})^2+16^2$
$\therefore$ AD = $\sqrt{105+256}=\sqrt{361}=19$ cm
Hence, the correct answer is 19 cm.
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