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

It is currently 15 Oct 2019, 22:43

Close

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
Your Progress

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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

In the figure, ABC is an equilateral triangle, and DAB is a right

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
Manager
Manager
avatar
Joined: 07 Feb 2010
Posts: 116
In the figure, ABC is an equilateral triangle, and DAB is a right  [#permalink]

Show Tags

New post 01 Dec 2010, 06:29
5
17
00:00
A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

53% (01:43) correct 47% (01:48) wrong based on 267 sessions

HideShow timer Statistics

Image
In the figure, ABC is an equilateral triangle, and DAB is a right triangle. What is the area of the circumscribed circle?

(1) DA = 4
(2) Angle ABD = 30 degrees

Attachment:
Untitled.pdf [10.85 KiB]
Downloaded 193 times
To download please login or register as a user

Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 58347
In the figure, ABC is an equilateral triangle, and DAB is a right  [#permalink]

Show Tags

New post 03 Mar 2012, 14:31
8
5
Image

In the figure, ABC is an equilateral triangle, and DAB is a right triangle. What is the area of the circumscribed circle?

You should know the following properties to solve this question:
• All inscribed angles that subtend the same arc are equal. The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle. Hence, all inscribed angles that subtend the same arc are equal.
• A right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle.
• In a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio \(1 : \sqrt{3}: 2\). Notice that the smallest side (1) is opposite the smallest angle (30°), and the longest side (2) is opposite the largest angle (90°).
For more check Circles Triangles and chapters of Math Book: http://gmatclub.com/forum/math-circles-87957.html and http://gmatclub.com/forum/math-triangles-87197.html

So, from above we'll have that as DAB=90 degrees then DB must be a diameter of the circle. Next, as angles ACB and ADB subtend the same arc AB then they must be equal and since ACB=60 (remeber ACB is an equilateral triangle) then ADB=60 too. Thus DAB is 30-60-90 triangle and its sides are in ratio \(1 : \sqrt{3}: 2\).

(1) DA = 4 --> the side opposite 30 degrees is 4, then hypotenuse DB=diameter=4*2=8 --> radius=4 --> \(area=\pi{r^2}=16\pi\). Sufficient.

(2) Angle ABD = 30 degrees --> we knew this from the stem, so nothing new. Not sufficient.

Answer: A.

Hope it's clear.

Attachment:
Inscribed Triangles.gif
Inscribed Triangles.gif [ 2.27 KiB | Viewed 18879 times ]

_________________
General Discussion
Intern
Intern
avatar
Joined: 03 Nov 2010
Posts: 11
In the figure, ABC is an equilateral triangle, and DAB is a right  [#permalink]

Show Tags

New post 03 Apr 2012, 16:46
I don't get why the Central Angle Theorem applies. In the figure, there is no central angle for that to occur with. So why would ADB be the same as ACB?
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 58347
In the figure, ABC is an equilateral triangle, and DAB is a right  [#permalink]

Show Tags

New post 04 Apr 2012, 01:15
1
KGG88 wrote:
I don't get why the Central Angle Theorem applies. In the figure, there is no central angle for that to occur with. So why would ADB be the same as ACB?


All inscribed angles that subtend the same arc are equal. Since angles ACB and ADB subtend the same arc AB then they must be equal. Next, there is no central angle shown on the diagram, I just mentioned Central Angle Theorem to explain why is above property true: angles ACB and ADB have the same central angle AOB (O is the center of the circle) and since they both equal to half of it then they must be equal.

For more check Circles and Triangles chapters of Math Book: http://gmatclub.com/forum/math-circles-87957.html and http://gmatclub.com/forum/math-triangles-87197.html
_________________
Intern
Intern
avatar
Joined: 26 Jul 2010
Posts: 22
Reviews Badge
Re: In the figure, ABC is an equilateral triangle, and DAB is a right  [#permalink]

Show Tags

New post 03 Apr 2013, 13:43
So I chose both statements together are sufficient but here is the explanation provided:
Since both angles ACB and ADB are opposite side AB, and angle ACB is part of an equilateral triangle, we can say both ACB and ADB are 60 degrees.

Why does that hold true? I get that :
1. DAB is 90
2. CAB = ACB = BAC = 60

and based on that all i can say is ABD < 60
so if we arent told ABD's value, cant ABD = 70 and ADB = 20?

In any case, if we can deduce ABD from just statement 2, then the rest is easy. if ABD = 30 degrees (as told by statement 1), then DB = diameter. we can figure out the length of DB using the 30-60-90 relationship and solve for area.

thanks.
VP
VP
User avatar
Status: Far, far away!
Joined: 02 Sep 2012
Posts: 1019
Location: Italy
Concentration: Finance, Entrepreneurship
GPA: 3.8
GMAT ToolKit User
Re: In the figure, ABC is an equilateral triangle, and DAB is a right  [#permalink]

Show Tags

New post 03 Apr 2013, 13:59
3
wown wrote:
So I chose both statements together are sufficient but here is the explanation provided:
Since both angles ACB and ADB are opposite side AB, and angle ACB is part of an equilateral triangle, we can say both ACB and ADB are 60 degrees.

Why does that hold true? I get that :
1. DAB is 90
2. CAB = ACB = BAC = 60

and based on that all i can say is ABD < 60
so if we arent told ABD's value, cant ABD = 70 and ADB = 20?

In any case, if we can deduce ABD from just statement 2, then the rest is easy. if ABD = 30 degrees (as told by statement 1), then DB = diameter. we can figure out the length of DB using the 30-60-90 relationship and solve for area.

thanks.


There is a thing you are not considering: in any right triangle inscribed in a circle, the hypotenuse coincides with a diameter of the circle.
So we know that DB is the diameter, that the bisectors of the ACB triangle intersect at the center (ancient math theory if I remember...).
ABC=60° is divided in two 30°angles, DAB=90° and so ADB=60°
And now as you said "we can figure out the length of DB using the 30-60-90 relationship and solve for area."
_________________
It is beyond a doubt that all our knowledge that begins with experience.
Kant , Critique of Pure Reason

Tips and tricks: Inequalities , Mixture | Review: MGMAT workshop
Strategy: SmartGMAT v1.0 | Questions: Verbal challenge SC I-II- CR New SC set out !! , My Quant

Rules for Posting in the Verbal Forum - Rules for Posting in the Quant Forum[/size][/color][/b]
Manager
Manager
avatar
Status: Looking to improve
Joined: 15 Jan 2013
Posts: 147
GMAT 1: 530 Q43 V20
GMAT 2: 560 Q42 V25
GMAT 3: 650 Q48 V31
Re: In the figure, ABC is an equilateral triangle, and DAB is a right  [#permalink]

Show Tags

New post 03 Apr 2013, 14:28
1
Wown,

This is very good trap C DS questions and if running out of time on exam you want to guess anything but C.

That said, the problem is using the relationship between an inscribed angle and central angle. central angle is the angle formed by an arc at the center and inscribed angle is the angle formed by the same arc at the circle.

central angle = 2 * inscribed angle - Here <ACB and <ADB are inscribed angles formed by arc AB.

From the stem, it is given that <ACB = 60 so the central angle for arch AB is 120. Make O as the center of the circle, so <AOB = 120

Now <ADB is the inscribed angle for <AOB so it will be 60 degrees. Triangle ADB is a 30-60-90 type right angle triangle and stmt 2 gives DA = 8 using which you can calculate the diameter of the circle as 16 and circumference as 16pi. Hence stmt B is sufficient.

Thanks for sharing this problem...

//Kudos please, if the above explanation is good.
_________________
KUDOS is a way to say Thank You
Verbal Forum Moderator
User avatar
B
Joined: 10 Oct 2012
Posts: 591
Re: In the figure, ABC is an equilateral triangle, and DAB is a right  [#permalink]

Show Tags

New post 03 Apr 2013, 21:40
wown wrote:
I came across a DS question whose explanation provides reasoning I do not follow. What am i missing?

Image


We know that angle subtended by the same arc, on the circumference are always equal.

From F.S 1, the data given is of no value. It is so because Angle ACB = Angle ADB = 60 degrees. Thus, angle ABD has to be (90-60) = 30 degrees.Insufficient.

From F.S 2, we know the length of AD. Thus, in the case of a 60-30-90 triangle, we can easily calculate the hypotenuse = the diameter =BD and hence the area.Sufficient.

B.
_________________
Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 13161
Re: In the figure (attached), ABC is an equilateral triangle,  [#permalink]

Show Tags

New post 22 Sep 2019, 00:44
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Club Bot
Re: In the figure (attached), ABC is an equilateral triangle,   [#permalink] 22 Sep 2019, 00:44
Display posts from previous: Sort by

In the figure, ABC is an equilateral triangle, and DAB is a right

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne