Question

In the given figure, XY is a tangent to the circle with centre O at A. If $\angle CAX=\angle BAY={60}^{\circ }$, then OD equals

• A.
  $\frac{1}{3}OA$
• B.
  $\frac{2}{3}OA$
• C.
  $DE$
• D.
  $\frac{2}{3}DE$

Answer 1

The correct option is C

$DE$
Given: XY is the tangent to the circle with centre O at A and $\angle CAX=\angle BAY={60}^{\circ }$

$\text{By Alternate Segment Theorem}\angle BCA=\angle BAY={60}^{\circ }Similarly,\angle ABC=\angle CAX={60}^{\circ }Therefore,\Delta ABC\text{is an equilateral triangle}\text{circumscribing the circle with centre O.}\text{Thus, centre O is also the centroid.}\Rightarrow OA:OD=2:1Also,OA=OE\left(Radii\right)\therefore OD=DE$

Hence the correct answer is option c.