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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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09 Sep 2018, 06:53
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93% (02:23) correct 7% (04:15) wrong based on 25 sessions

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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

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image024.jpg [ 2.4 KiB | Viewed 611 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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09 Sep 2018, 07:23
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

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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09 Sep 2018, 16:00
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 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?

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 $$\frac{3}{2}$$

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

*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   [#permalink] 09 Sep 2018, 16:00
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