Find all School-related info fast with the new School-Specific MBA Forum

It is currently 29 Jul 2016, 19:02
GMAT Club Tests

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.

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

Geometry: triangle props. Equilateral vs Isosceles, who's the biggest?

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

Hide Tags

Manager
Manager
User avatar
Joined: 08 Dec 2015
Posts: 203
GMAT 1: 600 Q44 V27
Followers: 1

Kudos [?]: 5 [0], given: 33

Reviews Badge
Geometry: triangle props. Equilateral vs Isosceles, who's the biggest? [#permalink]

Show Tags

New post 03 May 2016, 02:33
Hi everyone,

A question on triangle properties here.

From Gmatclub book:

For a given perimeter equilateral triangle has the largest area.
For a given area equilateral triangle has the smallest perimeter.


and

For an isosceles triangle with given length of equal sides right triangle (included angle) has the largest area.

So what is the biggest triangle? If we have to choose between a equilateral and an isosceles with the same perimeter, which one would have the biggest area?

Or these rules are not interrelated? I'm confused here..

Example when the equilateral rules is used:
is-the-perimeter-of-triangle-abc-greater-than-87112.html

Why don't we consider a right isosceles here? Why only the equilateral?

Example when the isosceles rule is used:
the-points-r-t-and-u-lie-on-a-circle-that-has-radius-4-if-168676.html

Then why not consider an equilateral here?..

I hope you get my doubt.

Thank you!
Expert Post
1 KUDOS received
Magoosh GMAT Instructor
User avatar
Joined: 28 Dec 2011
Posts: 3262
Followers: 1095

Kudos [?]: 4794 [1] , given: 53

Re: Geometry: triangle props. Equilateral vs Isosceles, who's the biggest? [#permalink]

Show Tags

New post 03 May 2016, 13:47
1
This post received
KUDOS
Expert's post
iliavko wrote:
Hi everyone,

A question on triangle properties here.

From Gmatclub book:

For a given perimeter equilateral triangle has the largest area.
For a given area equilateral triangle has the smallest perimeter.


and

For an isosceles triangle with given length of equal sides right triangle (included angle) has the largest area.

So what is the biggest triangle? If we have to choose between a equilateral and an isosceles with the same perimeter, which one would have the biggest area?

Or these rules are not interrelated? I'm confused here..

Example when the equilateral rules is used:
is-the-perimeter-of-triangle-abc-greater-than-87112.html

Why don't we consider a right isosceles here? Why only the equilateral?

Example when the isosceles rule is used:
the-points-r-t-and-u-lie-on-a-circle-that-has-radius-4-if-168676.html

Then why not consider an equilateral here?..

I hope you get my doubt.

Thank you!

Dear iliavko,
I'm happy to respond. :-) My friend, I believe you are confusing a few different things.

First of all, for the problem of the largest area with fixed perimeter, it depends very much on the starting criterion. Among all shapes in the plane, no restrictions, the shape with the most area for a fixed perimeter is the circle. If we limit ourselves to triangles, it's an equilateral triangle. If we limit ourselves to quadrilaterals, it's a square. If we limit it to all n-sided polygons for some specific value of n, it would be the regular polygon (notice that the equilateral triangle and the square are the regular polygons for the cases n = 3 and n = 4 respectively). This is mathematically true and relatively unlikely to be tested on the GMAT. Admittedly, it did play a role in the first question you cited:
is-the-perimeter-of-triangle-abc-greater-than-87112.html

Now, your quoted this rule.
For an isosceles triangle with given length of equal sides, the right triangle (included angle) has the largest area
Here, notice the nature of the constraint is very different. We are not fixing the entire perimeter, but instead, we are fixing the lengths of the two equal sides, and the third side can vary to be any possible length. This may seem like only a minor change, but it makes this a profoundly different question. In fact, this is not merely true for an isosceles triangle. I will generalize. Suppose segment AB has one fixed length and segment BC has another fixed length, and we construct triangle ABC; triangle ABC will have the maximum possible area when the angle at vertex B equals 90 degrees.
Attachment:
triangles with varying angle.JPG
triangles with varying angle.JPG [ 17.43 KiB | Viewed 230 times ]

This is a mathematical rule that is extremely different from the first rule. The nature of the question is quite different, so the nature of the answer is quite different.

Furthermore, in this question:
the-points-r-t-and-u-lie-on-a-circle-that-has-radius-4-if-168676.html
The solution involved an isosceles triangle only because it was known at the outset that two side of the triangle were radii, and hence the triangle must be isosceles. This is always the case for a triangle inside a circle that has two radii as two of its three sides. This has absolutely nothing to do with the rule I just discussed.

Does all this make sense?
Mike :-)
_________________

Mike McGarry
Magoosh Test Prep

Image

Image

Manager
Manager
User avatar
Joined: 08 Dec 2015
Posts: 203
GMAT 1: 600 Q44 V27
Followers: 1

Kudos [?]: 5 [0], given: 33

Reviews Badge
Re: Geometry: triangle props. Equilateral vs Isosceles, who's the biggest? [#permalink]

Show Tags

New post 06 May 2016, 09:16
Hi, Mike

Thank you for your reply! Yes, I get it now, thank you very much!

So just to make it clear, if for any reason we have to chose between an equilateral and an isosceles triangle for the
greatest are for a given perimeter, the equilateral wins. correct?

These relations between the shapes are very useful for ratio questions. if-the-perimeter-of-square-region-s-and-the-perimeter-of-cir-26871.html like here, if you know that the circle has the greatest area then you can take a 33% (more like a 50%) guess.

Thank you for the info!
Expert Post
Magoosh GMAT Instructor
User avatar
Joined: 28 Dec 2011
Posts: 3262
Followers: 1095

Kudos [?]: 4794 [0], given: 53

Re: Geometry: triangle props. Equilateral vs Isosceles, who's the biggest? [#permalink]

Show Tags

New post 06 May 2016, 15:25
Expert's post
iliavko wrote:
Hi, Mike

Thank you for your reply! Yes, I get it now, thank you very much!

So just to make it clear, if for any reason we have to chose between an equilateral and an isosceles triangle for the
greatest are for a given perimeter, the equilateral wins. correct?

These relations between the shapes are very useful for ratio questions. if-the-perimeter-of-square-region-s-and-the-perimeter-of-cir-26871.html like here, if you know that the circle has the greatest area then you can take a 33% (more like a 50%) guess.

Thank you for the info!

Dear iliavko,
I'm happy to respond. :-)

Yes, for a fixed perimeter, the triangle with the greatest area is the equilateral. That's mathematically true, but not likely to be all that helpful in GMAT math questions.

For the question you cited, this knowledge is less important, and knowing how to calculate the estimation is more important.

Does all this make sense?
Mike :-)
_________________

Mike McGarry
Magoosh Test Prep

Image

Image

Re: Geometry: triangle props. Equilateral vs Isosceles, who's the biggest?   [#permalink] 06 May 2016, 15:25
    Similar topics Author Replies Last post
Similar
Topics:
Experts publish their posts in the topic Geometry GMATD11 8 02 Feb 2011, 06:59
Experts publish their posts in the topic Geometry GMATD11 2 01 Feb 2011, 03:28
Area of an inscribed equilateral triangle? ksharma12 2 07 Apr 2010, 16:50
Geometry ? bamij 4 14 Jan 2010, 00:26
1 GEOMETRY - TRIANGLES amitdgr 11 17 Sep 2008, 23:02
Display posts from previous: Sort by

Geometry: triangle props. Equilateral vs Isosceles, who's the biggest?

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


GMAT Club MBA Forum Home| About| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group and phpBB SEO

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.