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

It is currently 18 Sep 2019, 19:25

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

A semicircle with area of xπ is marked by seven points equally spaced

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

Hide Tags

Find Similar Topics 
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 58092
A semicircle with area of xπ is marked by seven points equally spaced  [#permalink]

Show Tags

New post 19 Jul 2017, 23:30
1
16
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

41% (03:12) correct 59% (03:08) wrong based on 76 sessions

HideShow timer Statistics

A semicircle with area of \(x \pi\) is marked by seven points equally spaced along the half arc of the semicircle, such that two of the seven points form the endpoints of the diameter. What is the probability of forming a triangle with an area less than x from the total number of triangles formed by combining two of the seven points and the center of the diameter?

A. 4/5
B. 6/7
C. 17/21
D. 19/21
E. 31/35

_________________
Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 7831
A semicircle with area of xπ is marked by seven points equally spaced  [#permalink]

Show Tags

New post 20 Jul 2017, 06:40
5
2
Bunuel wrote:
A semicircle with area of \(x \pi\) is marked by seven points equally spaced along the half arc of the semicircle, such that two of the seven points form the endpoints of the diameter. What is the probability of forming a triangle with an area less than x from the total number of triangles formed by combining two of the seven points and the center of the diameter?

A. 4/5
B. 6/7
C. 17/21
D. 19/21
E. 31/35


HI,

Surely a 700 level Q...
BUT the choices can make it a sub 600 level Q...
if you know this much that the total triangle possible is 7C2-1=20

rest alll choices have multiple of 7 in denominator which is not possible..
ONLY A is left



firstly semi circle has a area of \(x \pi\)..
so \(\frac{\pi*r^2}{2}=x\pi.......r=\sqrt{2x}\)
so when we make 7 points in the way it has been described, there are 6 equal segments which will have area of \(\frac{x \pi}{6}\)..LESS than \(x\pi\)
here central angle at centre is 180/6=30..
How many triangles ? 6

Now when we take two such pieces, the centre angle becomes 60 and other two sides are radius so it becomes EQUILATERAL triangle with each side \(\sqrt{2x}\)..
Area = \(\sqrt{3}/4*a^2=\sqrt{3}/4*\sqrt{2x}^2\)=\(\sqrt{3}/2*x\), which is less than x.
How many such triangles? when 1 and 3 is choosen OR 2 and 4 OR 3 and 5 OR 4 and 6 OR 5 and 7------- 5 triangles..

Next when you choose three segments that is 30*3=90, it becomes right angled triangle with sides \(\sqrt{2x}\)..
area = \(\frac{1}{2}*\sqrt{2x}^2=x\) which is NOT less than x.
How many such triangles ? 1-4, 2-5,3-6,4-7 points -----4 such triangles possible

After this the central angle will become >90 and so area will again start reducing or will become less than x


so total triangles with area equal to x is 4
total triangles possible = 7C2=\(\frac{7!}{5!2!}\)=21
..
but it includes the two points 1 and 7 where these three points form a straight line DIAMETER.. so 21-1=20
And triangles with area LESS than x is 20-4=16

probability = \(\frac{16}{20}=\frac{4}{5}\)

A
_________________
General Discussion
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 58092
Re: A semicircle with area of xπ is marked by seven points equally spaced  [#permalink]

Show Tags

New post 20 Jul 2017, 06:44
chetan2u wrote:
Bunuel wrote:
A semicircle with area of \(x \pi\) is marked by seven points equally spaced along the half arc of the semicircle, such that two of the seven points form the endpoints of the diameter. What is the probability of forming a triangle with an area less than x from the total number of triangles formed by combining two of the seven points and the center of the diameter?

A. 4/5
B. 6/7
C. 17/21
D. 19/21
E. 31/35


HI,

Bunuel, pl relook into the choices given..


firstly semi circle has a area of \(x \pi\)..
so \(\frac{\pi*r^2}{2}=x\pi.......r=\sqrt{2x}\)
so when we make 7 points in the way it has been described, there are 6 equal segments which will have area of \(\frac{x \pi}{6}\)..LESS than \(x\pi\)
here central angle at centre is 180/6=30..
How many triangles ? 6

Now when we take two such pieces, the centre angle becomes 60 and other two sides are radius so it becomes EQUILATERAL triangle with each side \(\sqrt{2x}\)..
Area = \(\frac{\sqrt{3[}{square_root]/4}*a^2=\frac{[square_root]3[}{square_root]/4}*[square_root]2x}^2=\frac{\sqrt{3[}{square_root]/2}*x\), which is less than x.
How many such triangles? when 1 and 3 is choosen OR 2 and 4 OR 3 and 5 OR 4 and 6 OR 5 and 7------- 5 triangles..

Next when you choose three segments that is 30*3=90, it becomes right angled triangle with sides \([square_root]2x}\)..
area = \(\frac{1}{2}*\sqrt{2x}^2=x\) which is NOT less than x.
From here on any triangle with three segments or MORE will have AREA equal to or greater than x..

so total triangles with area less than x is 6+5=11
total triangles possible = 7C2=\(\frac{7!}{5!2!}\)=21
..
but it includes the two points 1 and 7 where these three points form a straight line DIAMETER.. so 21-1=20

probability = \(\frac{11}{20}\)


Checked. The options are copied as they show up in the source.
_________________
Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 7831
A semicircle with area of xπ is marked by seven points equally spaced  [#permalink]

Show Tags

New post 20 Jul 2017, 06:50
Bunuel wrote:
chetan2u wrote:
Bunuel wrote:
A semicircle with area of \(x \pi\) is marked by seven points equally spaced along the half arc of the semicircle, such that two of the seven points form the endpoints of the diameter. What is the probability of forming a triangle with an area less than x from the total number of triangles formed by combining two of the seven points and the center of the diameter?

A. 4/5
B. 6/7
C. 17/21
D. 19/21
E. 31/35


HI,

Bunuel, pl relook into the choices given..


firstly semi circle has a area of \(x \pi\)..
so \(\frac{\pi*r^2}{2}=x\pi.......r=\sqrt{2x}\)
so when we make 7 points in the way it has been described, there are 6 equal segments which will have area of \(\frac{x \pi}{6}\)..LESS than \(x\pi\)
here central angle at centre is 180/6=30..
How many triangles ? 6

Now when we take two such pieces, the centre angle becomes 60 and other two sides are radius so it becomes EQUILATERAL triangle with each side \(\sqrt{2x}\)..
Area = \(\frac{\sqrt{3[}{square_root]/4}*a^2=\frac{[square_root]3[}{square_root]/4}*[square_root]2x}^2=\frac{\sqrt{3[}{square_root]/2}*x\), which is less than x.
How many such triangles? when 1 and 3 is choosen OR 2 and 4 OR 3 and 5 OR 4 and 6 OR 5 and 7------- 5 triangles..

Next when you choose three segments that is 30*3=90, it becomes right angled triangle with sides \([square_root]2x}\)..
area = \(\frac{1}{2}*\sqrt{2x}^2=x\) which is NOT less than x.
From here on any triangle with three segments or MORE will have AREA equal to or greater than x..

so total triangles with area less than x is 6+5=11
total triangles possible = 7C2=\(\frac{7!}{5!2!}\)=21
..
but it includes the two points 1 and 7 where these three points form a straight line DIAMETER.. so 21-1=20

probability = \(\frac{11}{20}\)


Checked. The options are copied as they show up in the source.


Agreed Bunuel but it is surely 700 level Q
it skipped my mind that as the angle increases from 90 the area will again start reducing, so all other triangles will also be LESS than x
_________________
Intern
Intern
avatar
B
Joined: 22 Aug 2017
Posts: 3
Re: A semicircle with area of xπ is marked by seven points equally spaced  [#permalink]

Show Tags

New post 09 Aug 2018, 08:50
@Brunuel,@
Please explain this question with a diagram.
:(
Thank You
GMAT Club Bot
Re: A semicircle with area of xπ is marked by seven points equally spaced   [#permalink] 09 Aug 2018, 08:50
Display posts from previous: Sort by

A semicircle with area of xπ is marked by seven points equally spaced

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





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