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In the figure above, the measure of angle AOB is 60°. If the length of

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In the figure above, the measure of angle AOB is 60°. If the length of  [#permalink]

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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


Attachment:
image024.jpg
image024.jpg [ 2.4 KiB | Viewed 549 times ]

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In the figure above, the measure of angle AOB is 60°. If the length of  [#permalink]

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New post 09 Sep 2018, 07:23
Bunuel wrote:
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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


Attachment:
image024.jpg


length of the arc CD\(=2\pi*OC*(\frac{60}{360})=\frac{\pi}{3}*OC\)

\(OC=OA-AC=OA-\frac{OC}{2}\)
Or, \(OC=\frac{2}{3}*OA\)

Given, AB=27
Or, \(2\pi*OA*(\frac{60}{360})=27\)
Or, \(OA=\frac{81}{\pi}\)

So, Arc CD=\(\frac{\pi}{3}*OC=\frac{\pi}{3}*2/3*OA=\frac{\pi}{3}*\frac{2}{3}*\frac{81}{\pi}\)=18

Ans. (D)
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In the figure above, the measure of angle AOB is 60°. If the length of  [#permalink]

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New post 09 Sep 2018, 16:00
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Bunuel wrote:
Image
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 sectors
We 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 CD

Corresponding 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:

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

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

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

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

\(54=3(CD)\)
\(CD=18\)
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In the figure above, the measure of angle AOB is 60°. If the length of &nbs [#permalink] 09 Sep 2018, 16:00
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In the figure above, the measure of angle AOB is 60°. If the length of

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