honeyhani wrote:

In the figure below, AB is the chord of a circle with center O. AB is extended to C such that BC = OB. The straight line CO is produced to meet the circle at D. If ACD= y degrees and AOD = x degrees such that x = ky, then the value of k is

A. 3

B. 2

C. 1

D. None of the above

please help me with answers in details.

Sol:

OB = BC (given) => Angle BOC = Angle BCO = y => Angle OBC = 180-2y (in triangle OBC)

Hence Angle OBA = 2y

Since AO = OB (Both radius of the circle)

so Angle OBA = Angle BAO = 2y => Angle AOB = 180 -4y

Now we know all the three angles at point O form by a straight line DC

Hence Angle AOD + Angle AOB + Angle BOC = 180

x + 180 - 4y + y = 180

x - 3y = 0

x = 3y

Answer (A)