Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 24 Nov 2010
Posts: 6

Re: Perimeter of triangle ABC. [#permalink]
Show Tags
17 Dec 2010, 09:01
Bunuel wrote: sriharimurthy wrote: Thanks Bunuel.
Though I just realized something. I considered the area given to be 40 by mistake. It is actually 20.
Although the logic will be same, the calculations will be harder since we will have to use the real value of \(\sqrt{3}\). (no margin for approximations!)
Sorry for the mistake guys. However, as long as you understand the logic, it shouldn't matter. At least you'll know how to go about approaching such questions in the future!
Cheers.
Ps. Any trick for the calculations Bunuel? I would go backward. Let's assume the perimeter is 20. The largest area with given perimeter will have the equilateral triangle, so side=20/3. Let's calculate the area and if the area will be less than 20 it'll mean that perimeter must be more than 20. \(Area=s^2*\frac{\sqrt{3}}{4}=(\frac{20}{3})^2*\frac{\sqrt{3}}{4}=\frac{100*\sqrt{3}}{9}=~\frac{173}{9}<20\) Think this way is easier. \(\sqrt{3}\approx{1.73}\). Bunuel, is it necessary to solve it? what if you just recognized that you could solve it and the answer is either > 20 or < 20? I'm just thinking in terms of timing strategy...



Math Expert
Joined: 02 Sep 2009
Posts: 39702

Re: Perimeter of triangle ABC. [#permalink]
Show Tags
17 Dec 2010, 09:51
psirus wrote: Bunuel wrote: sriharimurthy wrote: Thanks Bunuel.
Though I just realized something. I considered the area given to be 40 by mistake. It is actually 20.
Although the logic will be same, the calculations will be harder since we will have to use the real value of \(\sqrt{3}\). (no margin for approximations!)
Sorry for the mistake guys. However, as long as you understand the logic, it shouldn't matter. At least you'll know how to go about approaching such questions in the future!
Cheers.
Ps. Any trick for the calculations Bunuel? I would go backward. Let's assume the perimeter is 20. The largest area with given perimeter will have the equilateral triangle, so side=20/3. Let's calculate the area and if the area will be less than 20 it'll mean that perimeter must be more than 20. \(Area=s^2*\frac{\sqrt{3}}{4}=(\frac{20}{3})^2*\frac{\sqrt{3}}{4}=\frac{100*\sqrt{3}}{9}=~\frac{173}{9}<20\) Think this way is easier. \(\sqrt{3}\approx{1.73}\). Bunuel, is it necessary to solve it? what if you just recognized that you could solve it and the answer is either > 20 or < 20? I'm just thinking in terms of timing strategy... You have to solve it. If we get that the minimum perimeter possible for a triangle with an area of 20 is less than 20 then we won't be able to answer the question. Similarly if we get that the maximum area possible for a triangle with a perimeter of 20 is more than 20 (for example 25) then knowing that area is 20 won't mean that perimeter must be more than 20. Hope it's clear.
_________________
New to the Math Forum? Please read this: All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



VP
Status: There is always something new !!
Affiliations: PMI,QAI Global,eXampleCG
Joined: 08 May 2009
Posts: 1326

Re: Perimeter of triangle ABC. [#permalink]
Show Tags
14 Jun 2011, 22:07
A bcac < ab < bc + ac ab > 10 and bc > 10 thus adding ab+bc = 20 already.Hence sufficient. B base * altitude = 40 base = bc altitude = ad now assuming values bc = 0.1,1 and 4 , ad = 400,40 and 10. all giving values of perimeter > 20. sufficient. D
_________________
Visit  http://www.sustainablesphere.com/ Promote Green Business,Sustainable Living and Green Earth !!



Intern
Joined: 28 Dec 2010
Posts: 23

Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
13 Jan 2012, 03:08
Hi Bunuel , Is there any way to find out the other two wxtreme in those two statement . Like : A. For a given perimeter equilateral triangle has the largest area: What is the lowest area ?
B. For a given area equilateral triangle has the smallest perimeter : What is the largest area ?



Manager
Joined: 29 Jul 2011
Posts: 107
Location: United States

Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
13 Jan 2012, 12:06
Good one: TRIANGLE RULES: 1. Third side < Sum of other two sides 2. Third side > Difference of other two sides 1. Starting with BC=11, AC=1, we get AB=9. Rule 1 is violated because BC > AB + AC. Keep on going BC = 12, 13... it gets worst. Therefore, the perimeter of the triangle has to be greater than 20. Suff. 2. This one I had to semiguess. I picked 345 triangle and found that area < perimeter. So, perimeter has to be > 20. Suff. D.
_________________
I am the master of my fate. I am the captain of my soul. Please consider giving +1 Kudos if deserved!
DS  If negative answer only, still sufficient. No need to find exact solution. PS  Always look at the answers first CR  Read the question stem first, hunt for conclusion SC  Meaning first, Grammar second RC  Mentally connect paragraphs as you proceed. Short = 2min, Long = 34 min



Intern
Joined: 22 May 2010
Posts: 34

Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
07 May 2012, 21:56
While solving statement2 in this problem, I realized that – For a right triangle with a given area, perimeter will be minimum when that right triangle is an "isosceles" right triangle.
I feel this property makes sense. Still, can someone confirm it please?
Using this property the problem can be solved as follows: Assume that the given right triangle with area 40, is an isosceles right triangle and then find the perimeter. If the perimeter of isosceles right triangle is greater than 20, that will mean any perimeter for that right triangle is definitely greater than 20.
Lets call base of right triangle = b Lets call height of right triangle = h Area given = 20. ==> (1/2)b.h = 20 ==> b.h = 40
If we assume this right triangle to be isosceles, then b=h= sq.root(40) => hypotenuse = sq.root(80) => perimeter = 2. sq.root(40) + sq.root(80) => This is greater than 20. => So minimum perimeter is more than 20. So any perimeter for this right triangle with the area as 20, will be more than 20. => Statement#2 is sufficient.



BSchool Forum Moderator
Joined: 27 Aug 2012
Posts: 1193

Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
29 Dec 2012, 08:21



Manager
Joined: 13 Oct 2012
Posts: 70
Concentration: General Management, Leadership

Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
03 Jan 2013, 22:46
very good question. thanks a lot for the explanation



Math Expert
Joined: 02 Sep 2009
Posts: 39702

Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
08 Jul 2013, 00:54



Intern
Joined: 05 May 2013
Posts: 27
GRE 1: 1480 Q800 V680

Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
09 Jul 2013, 09:55
(1). The Third side is greater than the difference betn the other two sides. Therefore atleast two sides are greater than 10. Therefore perimeter is >20. Sufficient.
(2) 1/2 bh =20 , if the base corresponding to this height is 1 (a smaller value gives an even greater height) , height is 40 , thus other side is greater > 40 (since other side is the hypotenuse in the right triangle formed by this height and base). Thus, perimeter > 20. Sufficient.
Answer is (D).



Senior Manager
Joined: 13 May 2013
Posts: 469

Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
10 Dec 2013, 06:26
Is the perimeter of triangle ABC greater than 20?
(1) BCAC=10. This can be rewritten as BC=10+AC. The sum of any two sides of a triangle must be greater than the third. So, for example, BC + AC must be greater than AB. Because BC is at least 10, AB must be greater than 10 as well. SUFFICIENT.
(2) The area of the triangle is 20. This I solved with a bit of intuition and drawing out but it's best to recognize that for triangles of the same area, perimeter is minimized when the triangle is equilateral. Knowing the area of an equilateral triangle will allow us to figure out a side length and thus, the perimeter: 20 = s^2 (Sqrt3/4) s^2 = 80/(Sqrt 3) s = 6.8 6.8+ 6.8+ 6.8 = 20.4 > 20
So when we solve for a triangle with the minimal possible value for it's given area it's greater than 20. SUFFICIENT.
D



Manager
Joined: 20 Oct 2013
Posts: 66

Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
10 May 2014, 12:30
Bunuel wrote: Is the perimeter of triangle ABC greater than 20?
(1) BCAC=10. (2) The area of the triangle is 20. Dear Bunnel as this was a yes no qs.... the ans was D... but if they had asked value of the perimeter... then the answer would be E right??
_________________
Hope to clear it this time!! GMAT 1: 540 Preparing again



Math Expert
Joined: 02 Sep 2009
Posts: 39702

Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
11 May 2014, 06:10



Intern
Joined: 15 Oct 2013
Posts: 29
Location: India
Concentration: General Management, Marketing
WE: Engineering (Energy and Utilities)

Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
20 Jun 2014, 03:15
An easier way to understand the calculation of the second statement is Following. Since for a given area an equilateral triangle has the smallest perimeter, Hence for an area of 20 , each side of the equilateral comes out to be 6.8. Hence the minimum perimeter for a triangle with area 20 is 3a=20.4; hence the perimeter has to be greater than 20. Hope it helps



GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15990

Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
28 Jun 2015, 07:19
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources



Intern
Joined: 06 Jul 2015
Posts: 8

Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
20 Sep 2015, 07:43
sriharimurthy wrote: Question Stem : Is AB + BC + AC > 20?
St. (1) : BC = AC + 10
Triangle Property : The sum of any two sides of a triangle is always greater than the third.
Since we are given that one of the sides is greater than 10, the sum of the other two sides must also be greater than 10. Hence the perimeter will always be greater than 20. Statement is sufficient.
St. (2) : A = 40
Triangle Property : For triangles with same area, the perimeter is smallest for an equilateral triangle.
Area of equilateral triangle with side x = \(\frac{\sqrt{3}}{4}x^2\)
Therefore, \(\frac{\sqrt{3}}{4}x^2\) = 40
\(x^2 = \frac{160}{\sqrt{3}}\)
Now, in order to speed up calculations, I will assume \(\sqrt{3}\) to be equal to 2.
If the condition is satisfied with \(\sqrt{3}\) equal to 2 then it will definitely be satisfied with the actual value of \(\sqrt{3}\) which is less than 2.
Therefore, \(x^2 = \frac{160}{2}\) = 80
This tells us that x is almost 9. More importantly, it tells us that x is greater than 8. Thus perimeter will be 3*x = 24.
Since this is the minimum perimeter possible (actually it is still less than what the actual minimum would be due to our approximations), we can conclude that the question stem will always be true.
Hence Sufficient.
Answer : D
Another interesting triangle property : For triangles with same perimeter, the area is maximum for an equilateral triangle. (If you think about it, this property goes hand in hand with the one we used in St. 2).
Dear Bunnel, Thanks for your explanation. I think I almost understood the logic but cannot figure out clearly how the triangle property you mentioned "For triangles with same perimeter, the area is maximum for an equilateral triangle" is related to the Statement 2. Can you please explain how I can link the property to the statement 2? Thanks in advanace regards Andy



Math Forum Moderator
Joined: 20 Mar 2014
Posts: 2643
Concentration: Finance, Strategy
GPA: 3.7
WE: Engineering (Aerospace and Defense)

Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
20 Sep 2015, 08:02
andy2whang wrote: sriharimurthy wrote: Question Stem : Is AB + BC + AC > 20?
St. (1) : BC = AC + 10
Triangle Property : The sum of any two sides of a triangle is always greater than the third.
Since we are given that one of the sides is greater than 10, the sum of the other two sides must also be greater than 10. Hence the perimeter will always be greater than 20. Statement is sufficient.
St. (2) : A = 40
Triangle Property : For triangles with same area, the perimeter is smallest for an equilateral triangle.
Area of equilateral triangle with side x = \(\frac{\sqrt{3}}{4}x^2\)
Therefore, \(\frac{\sqrt{3}}{4}x^2\) = 40
\(x^2 = \frac{160}{\sqrt{3}}\)
Now, in order to speed up calculations, I will assume \(\sqrt{3}\) to be equal to 2.
If the condition is satisfied with \(\sqrt{3}\) equal to 2 then it will definitely be satisfied with the actual value of \(\sqrt{3}\) which is less than 2.
Therefore, \(x^2 = \frac{160}{2}\) = 80
This tells us that x is almost 9. More importantly, it tells us that x is greater than 8. Thus perimeter will be 3*x = 24.
Since this is the minimum perimeter possible (actually it is still less than what the actual minimum would be due to our approximations), we can conclude that the question stem will always be true.
Hence Sufficient.
Answer : D
Another interesting triangle property : For triangles with same perimeter, the area is maximum for an equilateral triangle. (If you think about it, this property goes hand in hand with the one we used in St. 2).
Dear Bunnel, Thanks for your explanation. I think I almost understood the logic but cannot figure out clearly how the triangle property you mentioned "For triangles with same perimeter, the area is maximum for an equilateral triangle" is related to the Statement 2. Can you please explain how I can link the property to the statement 2? Thanks in advanace regards Andy You are not going to get a reply as the post you are quoting is from 2009. Let me try to answer your question. You are given a particular area in statement 2. Now, based on this value of area is there a property of triangles that you can apply to see whether you do get triangles with perimter >20 and area =20? As per the property mentioned, of all triangles with EQUAL areas, equilateral triangle will have the smallest perimeter. Thus, side of an equilateral triangle with area of 20 > \(\frac{\sqrt{3}*a^2}{4} = 20\), where a = side of the equilateral triangle. > a = (approx.) 6.79 > Perimeter (minimum of the triangles possible with area = 20 ) = 3*a=20.3 > 20. Thus, when the minimum perimeter is 20.3, then all the other possible traingles will have the perimeter > 20. Thus this statement is sufficient. Alternately, as mentioned by Bunuel at istheperimeteroftriangleabcgreaterthan87112.html#p836517, based on the property mentioned above, you know that for a given perimeter, an equilateral triangle will have the smallest area.Thus if the perimeter of the equilateral triangle is 20, then each side of the triangle = 20/3. Thus, the area of such an equilateral triangle = \(\frac{\sqrt{3}*a^2}{4} = 20\) = \(\frac{\sqrt{3}*(20/3)^2}{4} = =173/9 < 180/9 =20\). Thus we see that with perimeter 20 , the smallest area is <20. Thus, if we are given an area of 20 , then the perimeter of the smallest triangle (=an equilateral triangle) MUST be >20.
_________________
Thursday with Ron updated list as of July 1st, 2015: http://gmatclub.com/forum/consolidatedthursdaywithronlistforallthesections201006.html#p1544515 Rules for Posting in Quant Forums: http://gmatclub.com/forum/rulesforpostingpleasereadthisbeforeposting133935.html Writing Mathematical Formulae in your posts: http://gmatclub.com/forum/rulesforpostingpleasereadthisbeforeposting133935.html#p1096628 GMATCLUB Math Book: http://gmatclub.com/forum/gmatmathbookindownloadablepdfformat130609.html Everything Related to Inequalities: http://gmatclub.com/forum/inequalitiesmadeeasy206653.html#p1582891 Inequalities tips: http://gmatclub.com/forum/inequalitiestipsandhints175001.html Debrief, 650 to 750: http://gmatclub.com/forum/650to750a10monthjourneytothescore203190.html



Manager
Joined: 20 Apr 2014
Posts: 120

Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
27 Sep 2015, 04:33
For triangles with same area, the perimeter is smallest for an equilateral triangle. please Bunuel clarify this property. I do not grasp this concept so far. I memorize it but I can not apply it my self in similar question and when I should. thank you in advance for your great help.



Math Forum Moderator
Joined: 20 Mar 2014
Posts: 2643
Concentration: Finance, Strategy
GPA: 3.7
WE: Engineering (Aerospace and Defense)

Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
27 Sep 2015, 04:44
hatemnag wrote: For triangles with same area, the perimeter is smallest for an equilateral triangle. please Bunuel clarify this property. I do not grasp this concept so far. I memorize it but I can not apply it my self in similar question and when I should. thank you in advance for your great help. The concept IS what you mentioned. For all the triangles given to you with a fixed area, the equilateral triangle will have the smallest perimeter. This is a very unique property of equilateral triangles. Let's say in a PS question, you are given that area of a triangle is 20 square units, what is the smallest perimeter of this triangle ? You will be able to use the property above to see that for a triangle with a given area to have the smallest perimeter, it must be an equilateral triangle. Based on this you can use the formula \(\frac{\sqrt{3}*a^2}{4} = 20\) > calculate 'a' and then for perimeter= 3a This is how you will be able to apply this property. Hope this helps
_________________
Thursday with Ron updated list as of July 1st, 2015: http://gmatclub.com/forum/consolidatedthursdaywithronlistforallthesections201006.html#p1544515 Rules for Posting in Quant Forums: http://gmatclub.com/forum/rulesforpostingpleasereadthisbeforeposting133935.html Writing Mathematical Formulae in your posts: http://gmatclub.com/forum/rulesforpostingpleasereadthisbeforeposting133935.html#p1096628 GMATCLUB Math Book: http://gmatclub.com/forum/gmatmathbookindownloadablepdfformat130609.html Everything Related to Inequalities: http://gmatclub.com/forum/inequalitiesmadeeasy206653.html#p1582891 Inequalities tips: http://gmatclub.com/forum/inequalitiestipsandhints175001.html Debrief, 650 to 750: http://gmatclub.com/forum/650to750a10monthjourneytothescore203190.html



Manager
Joined: 06 Jun 2013
Posts: 128
Location: India
Concentration: Finance, Economics
GPA: 3.6
WE: Engineering (Computer Software)

Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink]
Show Tags
30 Sep 2015, 08:52
i got answer using another approach which may not be correct always. statement 2 :
1/2 * b * h = 20 b*h =40 possible combinations 10*4,20*2, 8*5 etc. in these combinations let's consider base as 10,20,8. and we know the angle property that sum of 2 sides is greater than the 3rd side, so in the above combinations sum of other 2 sides must be greater than 10 and 20. in 3rd case since height is 5, using common sense we can conclude that sum of other two sides must be greater than 12.




Re: Is the perimeter of triangle ABC greater than 20? (1)
[#permalink]
30 Sep 2015, 08:52



Go to page
Previous
1 2 3
Next
[ 41 posts ]




