Geometry: Perimeter of a triangle vs that of a square

This topic has expert replies
Master | Next Rank: 500 Posts
Posts: 103
Joined: Sat Jun 02, 2012 9:46 pm
Thanked: 1 times
Can someone please explain how you'd go about tackling this question?

Question: Is the perimeter of a triangle T greater than the perimeter of square S?

1) The length of the longest side of T is twice the length of a side S.

2) T is isoceles.

I assume you start by picking a variable for either the side of S or T. How do you quickly deduce from there?

the answer is

A

Senior | Next Rank: 100 Posts
Posts: 65
Joined: Wed Mar 28, 2012 6:54 pm
Location: Canada
Thanked: 15 times
Followed by:1 members

by Param800 » Thu Jan 03, 2013 8:13 pm
I did this way,

We know that " the sum of two sides of a triangle are always greater than the third side."

So, let the length of first, second and third sides of triangle be x, y and z respectively where z is the longest side.

So, we know that x + y >= z ... (i)

And for perimeter ... x + y + z >= 2 z ( from i )

OK Now, lets go to options..

St. 1 -- It says that the longest side is 2*a ( where a is the side of square )
So.. from above analysis... we can say that permeter of triangle will be >= 2 ( 2 *a) >= 4*a

and we know that perimeter of square is 4*a...so this statement is sufficient.

St2 -- It does not say anything about square, so insufficient.

Hence, answer is A. Hope this clarifies your concern
topspin360 wrote:Can someone please explain how you'd go about tackling this question?

Question: Is the perimeter of a triangle T greater than the perimeter of square S?

1) The length of the longest side of T is twice the length of a side S.

2) T is isoceles.

I assume you start by picking a variable for either the side of S or T. How do you quickly deduce from there?

the answer is

A

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 273
Joined: Tue Sep 21, 2010 5:37 am
Location: Raleigh, NC
Thanked: 154 times
Followed by:74 members
GMAT Score:770

by Whitney Garner » Thu Jan 03, 2013 8:15 pm
Hi topspin360!

This is a tricky problem to do algebraically because of the various configurations for the triangle (is it right, isosceles, equilateral, nothing??). So maybe we just start making up some numbers that work. We are only asked about perimeters, so as long as we make the sides of the triangle "legal" (no side is longer than the sum of the other 2 sides), then we're good.

(1) The length of the longest side of T is twice the length of a side S.
To be lazy here, I picked a well known right triangle: 3-4-5, perimeter=12. This means the side of the square is 1/2 of 5, or 2.5, perimeter=10. In this case, the perimeter of the triangle is LARGER than the perimeter of the square. Now I have to test something where the perimeter of the triangle would be LESS than the perimeter of the square....maybe if I make the sides of the triangle really small, and if I make the "longest" side as short as possible, or the same length as the other sides? What about an equilateral with all sides 1, perimeter=3. The square would have sides .5, perimeter 2...still smaller!! So let's think theory... the smallest "long" side for a triangle would be one where it was the same length as the other sides (equilateral), so if the triangle has side lengths t, perimeter 3t, the square would have sides .5T, perimeter 2t...always smaller than 3t!! So it looks like we were able to prove that under this constraint, the perimeter of the triangle is ALWAYS larger than that of the square! SUFFICIENT

(2) T is isoceles.
So here, we don't have ANY relationship between the triangle and the square so we can make up whatever we want. Let's make the triangle 1-1-1 (because ALL equilateral triangles are also isosceles), and the square 2-2-2-2... Here the square has a LARGER perimeter than the triangle. But swap them, make the triangle 2-2-2 and the square 1-1-1-1... Now the square is only 4 while the triangle is 6, so the triangle has a LARGER perimeter. NOT SUFFICIENT

Therefore, the answer is A

Hope this helps!
:)
Whit
Whitney Garner
GMAT Instructor & Instructor Developer
Manhattan Prep

Contributor to Beat The GMAT!

Math is a lot like love - a simple idea that can easily get complicated :)

User avatar
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

by GMATGuruNY » Thu Jan 03, 2013 11:55 pm
topspin360 wrote:Can someone please explain how you'd go about tackling this question?

Question: Is the perimeter of a triangle T greater than the perimeter of square S?

1) The length of the longest side of T is twice the length of a side S.

2) T is isoceles.

I assume you start by picking a variable for either the side of S or T. How do you quickly deduce from there?

the answer is

A
Statement 1: The length of the longest side of T is twice the length of a side pf S.
Square S:
Let each side = s.
Then the perimeter = 4s.

Triangle T:
Let triangle T be ∆ABC, whose longest side is AC.
Then AC = 2s.
Since the third side of a triangle must be LESS than the sum of the lengths of the other 2 sides, AC < AB+BC.
Thus, AB+BC > AC, implying that AB+BC > 2s.
Since AC = 2s, and AB+BC > 2s, AB+BC+AC > 4s.
In other words, the perimeter of triangle T is GREATER than 4s.

Since the perimeter of triangle T > 4s, and the perimeter of square S = 4s, the perimeter of triangle T is greater than the perimeter of square S.
SUFFICIENT.

Statement 2: T is isosceles.
No information about square S.
INSUFFICIENT.

The correct answer is A.
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