O is a point in the interior of ΔABC such that OA = 12 cm, OC = 9 cm, ∠AOB = ∠BOC = ∠COA and ∠ABC = 60°. What is the length (in cm) of OB?

Answer: Option A

Solution

Answer: Option A

Solution:

$$\begin{array}{rl}& \text{In }\Delta AOB\\ & \frac{\sin \theta }{OB}=\frac{\sin \left({60}^{\circ }-\theta \right)}{AO}\\ & \frac{\sin \theta }{\sin \left({60}^{\circ }-\theta \right)}=\frac{OB}{12}\text{. }\text{. }\text{. }\text{. }\text{. }\left(\text{i}\right)\\ & \text{In }\Delta BOC\\ & \frac{\sin \theta }{OC}=\frac{\sin \left({60}^{\circ }-\theta \right)}{OB}\\ & \frac{\sin \theta }{\sin \left({60}^{\circ }-\theta \right)}=\frac{9}{OB}\text{. }\text{. }\text{. }\text{. }\text{. }\left(\text{ii}\right)\\ & \text{Compare }\left(\text{i}\right)\text{ and }\left(\text{ii}\right)\\ & \frac{OB}{12}=\frac{9}{OB}\\ & O{B}^2=12\times 9\\ & O{B}^2=108\\ & OB=6\sqrt{3}\end{array}$$