Radii OA, OB, and OC divide the circle into three congruent parts. Let D be a point on the arc AB such that arc AD has half the measure of arc DB. Find the measure of angle OCD.

Guest Jun 13, 2020

#1**0 **

OA, OB, OC divide the circle into three congruent parts \(\Longleftrightarrow\) \(\angle AOB = \angle BOC = \angle COA = 120^\circ\)

D is a point on arc AB such that arc AD has half the measure of arc DB \(\Longleftrightarrow\) \(m\stackrel{\mbox{$\large\frown$}}{AD} = 40^\circ\), \(m\stackrel{\mbox{$\large\frown$}}{DB} = 80^\circ\)

\(\text{reflex }\angle COD = m\stackrel{\mbox{$\large\frown$}}{DB} + m\stackrel{\mbox{$\large\frown$}}{BC} = 200^\circ\)

\(m\angle COD = 360^\circ - 200^\circ = 160^\circ\)

Because OC = OD,

\(m\angle OCD = m\angle ODC\)

So \(160^\circ + 2 \left(m\angle OCD\right) = 180^\circ\)

I believe you can continue from here. It's just one step away from the answer.

MaxWong Jun 13, 2020