question_answer

A hoop is resting vertically at staircase as shown in the diagram. AB=12 cm and BC=8 cm. The radius of the hoop is _____.

A)  13 cm

B)  \[\text{12}\sqrt{2}\,\,cm\]

C)  14 cm

D)  \[\text{13}\sqrt{2}\,\,cm\]

Correct Answer: A

Solution :

\[\Delta OEC\] is a right angle triangle, right angled at E. Let \[\text{OA=OC}\] \[=\text{ }x\text{ }cm\text{ }(radius)\] Now, \[EC\text{ }=12\text{ }cm\] \[\text{AE = 8 cm}\] \[\therefore \]Now in \[\Delta OEC,\] \[O{{C}^{2}}=O{{E}^{2}}+E{{C}^{2}}\] \[\Rightarrow {{x}^{2}}={{(x-8)}^{2}}+{{12}^{2}}\] \[\Rightarrow {{x}^{2}}={{x}^{2}}+64-16x+144\] \[\Rightarrow 16x=208\Rightarrow x=13cm\]