GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 20 Feb 2019, 19:41

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Events & Promotions

Events & Promotions in February
PrevNext
SuMoTuWeThFrSa
272829303112
3456789
10111213141516
17181920212223
242526272812
Open Detailed Calendar
• Free GMAT Prep Hour

February 20, 2019

February 20, 2019

08:00 PM EST

09:00 PM EST

Strategies and techniques for approaching featured GMAT topics. Wednesday, February 20th at 8 PM EST

February 21, 2019

February 21, 2019

10:00 PM PST

11:00 PM PST

Kick off your 2019 GMAT prep with a free 7-day boot camp that includes free online lessons, webinars, and a full GMAT course access. Limited for the first 99 registrants! Feb. 21st until the 27th.

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

Author Message
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 53020
In the figure above, the measure of angle AOB is 60°. If the length of  [#permalink]

Show Tags

09 Sep 2018, 06:53
00:00

Difficulty:

25% (medium)

Question Stats:

93% (02:23) correct 7% (04:15) wrong based on 25 sessions

HideShow timer Statistics

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 [ 2.4 KiB | Viewed 611 times ]

_________________
Director
Status: Learning stage
Joined: 01 Oct 2017
Posts: 958
WE: Supply Chain Management (Energy and Utilities)
In the figure above, the measure of angle AOB is 60°. If the length of  [#permalink]

Show Tags

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)
_________________

Regards,

PKN

Rise above the storm, you will find the sunshine

Senior SC Moderator
Joined: 22 May 2016
Posts: 2489
In the figure above, the measure of angle AOB is 60°. If the length of  [#permalink]

Show Tags

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:

$$\frac{r_1}{s_1}=\frac{R_2}{S_2}$$

$$\frac{R_2}{r_1}=\frac{S_2}{s_1}$$

$$\frac{OA}{OC}=\frac{AB}{CD}$$

$$\frac{3}{2}=\frac{27}{CD}$$

$$54=3(CD)$$
$$CD=18$$
_________________

To live is the rarest thing in the world.
Most people just exist.

Oscar Wilde

In the figure above, the measure of angle AOB is 60°. If the length of   [#permalink] 09 Sep 2018, 16:00
Display posts from previous: Sort by