Last visit was: 22 Apr 2026, 15:31 It is currently 22 Apr 2026, 15:31
Home
Messages
Tests
Schools
Events
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.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 22 Apr 2026
Posts: 109,754
Own Kudos:
810,664
 [26]
Given Kudos: 105,823
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,754
Kudos: 810,664
 [26]
A semicircle with area of xπ is marked by seven points equally spaced 19 Jul 2017, 23:30
1
Kudos
Add Kudos
25
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

95% (hard)

Question Stats:

35% (03:02) correct 65% (02:50) wrong based on 153 sessions

History

Date
Time
Result
Not Attempted Yet
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
ShowHide Answer
Official Answer
A
Most Helpful Reply
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 22 Apr 2026
Posts: 11,229
Own Kudos:
44,994
 [12]
Given Kudos: 335
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,229
Kudos: 44,994
 [12]
A semicircle with area of xπ is marked by seven points equally spaced 20 Jul 2017, 06:40
8
Kudos
Add Kudos
4
Bookmarks
Bookmark this Post
Bunuel
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
Show more

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
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 22 Apr 2026
Posts: 109,754
Own Kudos:
810,664
Given Kudos: 105,823
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,754
Kudos: 810,664
A semicircle with area of xπ is marked by seven points equally spaced 20 Jul 2017, 06:44
Kudos
Add Kudos
Bookmarks
Bookmark this Post
chetan2u
Bunuel
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 = \(\frac{\sqrt{3}}{4}*a^2=\frac{\sqrt{3}}{4}*\sqrt{2x}^2=\frac{\sqrt{3}}{2}*x\), 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.
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 = \(\frac{11}{20}\)
Show more

Checked. The options are copied as they show up in the source.
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 22 Apr 2026
Posts: 11,229
Own Kudos:
44,994
Given Kudos: 335
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,229
Kudos: 44,994
A semicircle with area of xπ is marked by seven points equally spaced 20 Jul 2017, 06:50
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
chetan2u
Bunuel
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 = \(\frac{\sqrt{3}}{4}*a^2=\frac{\sqrt{3}}{4}*\sqrt{2x}^2=\frac{\sqrt{3}}{2}*x\), 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.
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.
Show more

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
User avatar
Goldenfuture
User avatar
Joined: 24 Dec 2024
Last visit: 29 Jan 2026
Posts: 150
Own Kudos:
12
Given Kudos: 48
Posts: 150
Kudos: 12
A semicircle with area of xπ is marked by seven points equally spaced 20 Dec 2025, 07:27
Kudos
Add Kudos
Bookmarks
Bookmark this Post
KarishmaB - Can you please share a simple answer here
User avatar
ZahraB
User avatar
Joined: 23 Dec 2022
Last visit: 03 Apr 2026
Posts: 6
Given Kudos: 3
Posts: 6
Kudos: 0
A semicircle with area of xπ is marked by seven points equally spaced 20 Dec 2025, 11:07
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hi there, please confirm if my method is correct:
Step 1: 7C3 = 7*6*5 divided by 3*2*1 = 35 (denominator) [ways of selecting 3 points from 7 points]
Step 2: For numerator: given 7 points lie on the same line (i.e., half arc) = 35 less 7 (to remove collinear points) = 28 [numerator]
Step 3: = 28 / 35 = 4/5 (option A).

Not sure about step 2 but got the right answer. Please someone check and confirm. thanks.
User avatar
KarishmaB
User avatar
Joined: 16 Oct 2010
Last visit: 21 Apr 2026
Posts: 16,439
Own Kudos:
79,387
Given Kudos: 484
Location: Pune, India
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 16,439
Kudos: 79,387
A semicircle with area of xπ is marked by seven points equally spaced 20 Dec 2025, 22:14
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Goldenfuture
KarishmaB - Can you please share a simple answer here
Show more

The question is not relevant for GMAT Focus. It expects you to understand that the maximum area will be when the triangle is a right triangle. I discussed this is an old video that I made for GMAT Classic here: https://www.youtube.com/watch?v=y05XIAWgAT0

The probability part is fairly straight forward but this understanding is not tested anymore. If you do understand this, then it is quick to solve.

Area of semi circle \(= \pi*r^2/2 = \pi * x\)
\(r = \sqrt{2x}\)

The total number of triangles made = 7C2 - 1 = 20 (Select any 2 points of the 7. They will all give triangle except one case in which the end points of the diameter are selected.)

The total number of right triangles will be 4.
Area of a right triangle \(= (1/2) * \sqrt{2x} * \sqrt{2x} = x\)


Area of all other triangles i.e. 20 - 4 = 16 triangles will be less than x.

Probability = 16/20 = 4/5

Answer (A)
Attachments

Screenshot 2025-12-21 at 10.44.17 AM.png
Screenshot 2025-12-21 at 10.44.17 AM.png [ 30.85 KiB | Viewed 491 times ]

Moderators:
Bunuel
Math Expert
109754 posts
Krunaal
Tuck School Moderator
853 posts