Equilateral triangle inscribed in a circle
This topic has expert replies
- Elena Plescan
- Junior | Next Rank: 30 Posts
- Posts: 18
- Joined: Tue Sep 16, 2014 10:14 am
- GMATGuruNY
- GMAT Instructor
- Posts: 15539
- Joined: Tue May 25, 2010 12:04 pm
- Location: New York, NY
- Thanked: 13060 times
- Followed by:1906 members
- GMAT Score:790
A central angle is formed by two radii.Equilateral triangle ABC is inscribed in a circle (points ABC are on the circle). IF the length of arc ABC is 24, what is the approximate diameter of the circle
A) 5
B) 8
C) 11
D) 15
E) 19
An inscribed angle is formed by two chords.
When an inscribed angle and a central angle intercept the same arc on the circle, the degree measurement of the inscribed angle is 1/2 the degree measurement of the central angle:
Circles display the following proportionality:
(Central Angle)/360 = (intercepted arc length)/circumference = (sector area)/(circle area)
Since 120/360 = 1/3, the intercepted arc in the circle above is 1/3 the circumference of the circle. The sector enclosed by the two radii is 1/3 the area of the entire circle.
Now here's a drawing of the problem above:
Let c = circumference.
Since angle A is 60 degrees, the corresponding central angle is 120 degrees. Since 120/360 = 1/3, arc BC = (1/3)c.
Using similar logic, arc AB = (1/3)c.
Thus, arc ABC = (2/3)c.
Since arc ABC = 24:
24 = 2/3c
c = 36.
Thus:
Ï€d = 36.
d ≈ 36/π = a little less than 12.
The correct answer is C.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
GMAT/MBA Expert
- Brent@GMATPrepNow
- GMAT Instructor
- Posts: 16207
- Joined: Mon Dec 08, 2008 6:26 pm
- Location: Vancouver, BC
- Thanked: 5254 times
- Followed by:1268 members
- GMAT Score:770
You're right Elena - the equilateral triangle cuts the circle in 3 equal arcs.Equilateral triangle ABC is inscribed in a circle (points ABC are on the circle). IF the length of arc ABC is 24, what is the approximate diameter of the circle
A) 5
B) 8
C) 11
D) 15
E) 19
However, arc ABC travels the length of 2 of those arcs (not 1).
So, each arc must have length 12, which means the TOTAL CIRCUMFERENCE = (3)(12) = 36
Now, we'll use the formula: CIRCUMFERENCE = (pi)(diameter)
So, 36 = (3.14)(diameter)
This means that: diameter = 36/3.14
IMPORTANT: We need not perform any long division here. Notice that the answer choices are nicely spread apart. So, we can ESTIMATE.
We know that 36/3 = 12
Since 3.14 is a bit bigger than 3, we know that 36/3.14 will be a bit smaller than 12.
Answer choice C is a bit smaller than 12, so it must be the correct answer.
Cheers,
Brent
- Elena Plescan
- Junior | Next Rank: 30 Posts
- Posts: 18
- Joined: Tue Sep 16, 2014 10:14 am
Hi Brent,
It's all much clearer now. I guess I just misunderstood the notation of the arc and didn't realize that arc ABC=arc AB + arc BC.
Thanks a lot,
Elena
It's all much clearer now. I guess I just misunderstood the notation of the arc and didn't realize that arc ABC=arc AB + arc BC.
Thanks a lot,
Elena