Bunuel wrote:

In the figure above, the measure of angle AOB is 60°. If the length of the arc AB is 27 and the length of AC is half of that of OC, what is the length of the arc CD?

A. 9

B. 12

C. 16

D. 18

E. 24

Use the concept of similar sectors and scale factor.

Similar sectorsWe want to use "similar sectors" because

• corresponding parts of similar figures are in proportion ("parts" such as radii and arc lengths)

• radius and arc length are related

• we have information about the radii that allows us to calculate the numeric relationship between them, AND

• If we find THAT relationship, it will apply to arc lengths.

All circles are similar.

Circle SECTORS with congruent (equal) central angles are similar.

Sectors AOB and COD both have a central angle of 60° and thus are similar.

Relationship between short and long radius?• Calculated from given information about radii OC and OA:

The radius of the small circle multiplied by \(\frac{3}{2}\) equals the radius of the large circle•

How to calculate that relationship

\(OC=r_1=\) length of radius of the small circle

\(OA=R_2=\) length of the radius of the large circle

\(AC = \frac{1}{2}OC\)

\(OC+AC=OA\)

\(OC+\frac{1}{2}OC=OA\)

\(\frac{3}{2}OC=OA\)

\(\frac{3}{2}*r_1=R_2\)

Thus the radius of the small circle, dilated by a scale factor of \(\frac{3}{2}\), equals the radius of the large circle

Find arc length CDCorresponding parts of similar figures are in proportion.*

Arc length is directly proportional to radius.

Or: radius length determines circumference. Arc length is a portion of circumference. Whatever happens to the radius will happen to circumference and any portion of circumference.

So the arc length of the smaller circle will also be dilated by a scale factor of \(\frac{3}{2}\)

\(CD*\frac{3}{2}=AB\)

\(CD=\frac{2}{3}*AB\)

\(CD=(\frac{2}{3}*27)=18\)

Answer D

*Arc length = \(s\), and corresponding parts of similar figures are in proportion, so:

\(\frac{r_1}{s_1}=\frac{R_2}{S_2}\)

\(\frac{R_2}{r_1}=\frac{S_2}{s_1}\)

\(\frac{OA}{OC}=\frac{AB}{CD}\)

\(\frac{3}{2}=\frac{27}{CD}\)

\(54=3(CD)\)

\(CD=18\)

_________________

In the depths of winter, I finally learned

that within me there lay an invincible summer.

-- Albert Camus, "Return to Tipasa"