Question : O is a point in the interior of $\triangle \mathrm{ABC}$ such that OA = 12 cm, OC = 9 cm, $\angle A O B=\angle B O C=\angle C O A$ and $\angle A B C=60^{\circ}$. What is the length (in cm) of OB?
Option 1: $6 \sqrt{3}$
Option 2: $4 \sqrt{6}$
Option 3: $4 \sqrt{3}$
Option 4: $6 \sqrt{2}$
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Correct Answer: $6 \sqrt{3}$
Solution :
O is a point in the interior of $\triangle \mathrm{ABC}$ OA = 12 cm OC = 9 cm $\angle A O B=\angle B O C=\angle C O A$ $\angle A B C=60^{\circ}$ $\angle A O B=\angle B O C=\angle C O A=\frac{360^{\circ}}{3} = 120^{\circ}$ Let $\angle OBA = \theta$ In $\triangle AOB$, $\angle OBA + \angle BAO + \angle AOB = 180^{\circ}$ ⇒ $\theta + \angle BAO + 120^{\circ} = 180^{\circ}$ ⇒ $\angle BAO = 60^{\circ} - \theta$ In $\triangle BOC$, $\angle OBC = \angle ABC - \angle OBA$ ⇒ $ \angle OBC = 60^{\circ} - \theta$ ⇒ $\angle OBC + \angle BOC + \angle OCB = 180^{\circ}$ ⇒ $60^{\circ} - \theta + 120^{\circ} + \angle OCB = 180^{\circ}$ ⇒ $\angle OCB = 180^{\circ}-120^{\circ}-60^{\circ}-\theta$ ⇒ $\angle OCB = \theta$ In $\triangle AOB$ and $\triangle BOC$, $\angle A = \angle B$ $\angle O = \angle O$ $\angle B = \angle C$ By AAA similarity, $\triangle AOB$ ~ $\triangle BOC$ ⇒ $\frac{OA}{OB} = \frac{OB}{OC}$ ⇒ $\frac{12}{OB} = \frac{OB}{9}$ ⇒ $OB^2 = 12 × 9$ ⇒ $OB = \sqrt{108} = 6\sqrt3$ Hence, the correct answer is $6\sqrt3$.
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