AB is a chord in a circle with centre O. AB is produced to C such that BC is equal to the radius of the circle. C is joined to O and produced to meet the circle at D. If ∠ACD = 32°, then the measure of ∠AOD is ______.
Answer & explanation
Correct answer: option 2
In \(\Delta \)OBC
If OB = BC, then
\(\angle\)BOC = \(\angle\)BCO = \({32}^\circ\)
As we know,
\(\angle\)OBA = \(\angle\)BOC + \(\angle\)BCO = 32 + 32 = 64
If OA = OB, then
\(\angle\)OBA = \(\angle\)OAB = 64
In\(\Delta \)AOB
\(\angle\)AOB + \(\angle\)OAB + \(\angle\)OBA = 180
\(\angle\)AOB + 64 + 64 = 180
\(\angle\)AOB = 180 - 128 = 52
\(\angle\)AOD + \(\angle\)AOB + \(\angle\)BOC = 180
\(\angle\)AOD + 52 + 32 = 180
\(\angle\)AOD = 180 - 84 = \({96}^\circ\).