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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