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# Which of the following is a possible length for side AB of triangle AB

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Director
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Which of the following is a possible length for side AB of triangle AB  [#permalink]

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Updated on: 17 Sep 2014, 00:40
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Which of the following is a possible length for side AB of triangle ABC if AC = 6 and BC = 9?

I. 3
II. 9 √3
III. 13.5

(A) I only
(B) II only
(C) III only
(D) II and III
(E) I, II, and III

Originally posted by jimmyjamesdonkey on 21 May 2008, 09:25.
Last edited by Bunuel on 17 Sep 2014, 00:40, edited 1 time in total.
Renamed the topic, edited the question, added the OA and moved to PS forum.
Director
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Re: Which of the following is a possible length for side AB of triangle AB  [#permalink]

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21 May 2008, 10:17
Should be II and III.
Sum to two sides of triangle should be greater than third side.
I is not possible because 3 + 6 = 9 which is not greater than 9 (length of third side).
For Both II and III you add any two pairs and the sum of pairs is greater than the length of third side.
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Re: Which of the following is a possible length for side AB of triangle AB  [#permalink]

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05 Jun 2008, 11:33
1
The answer is III.

It cant be II because 9*3^(0.5)>15, and the thirs side must be less than 15.
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Re: Which of the following is a possible length for side AB of triangle AB  [#permalink]

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05 Jun 2008, 14:05
The third side of a triangle should be less than the sum of the other 2 sides and greater than the mod(difference of 2 sides)

so, x should lie between mod(y-z)<x<y+z
(9-6)<x<9+6
3<x<15

only value is 13.5
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Re: Which of the following is a possible length for side AB of triangle AB  [#permalink]

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05 Jun 2008, 15:10
2 and 3 for me as well .... we need the third side to be between 3 and 15, i.e. 3<x<15

Both 2 and 3 fit this requirement
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Re: Which of the following is a possible length for side AB of triangle AB  [#permalink]

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05 Jun 2008, 15:34
9(3^1/2) = 15.58>15 ...only 3.
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Re: Which of the following is a possible length for side AB of triangle AB  [#permalink]

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05 Jun 2008, 16:01
yes, how silly of me
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Re: Which of the following is a possible length for side AB of triangle AB  [#permalink]

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05 Jun 2008, 23:54
Rules for a triangle
(1)Third side is always greater than the difference of two other sides
(2) Third side is always less than the sum of two other sides
Going by above rules only 13.5 satisfies the condition.
jimmyjamesdonkey wrote:
Which of the following is a possible length of the side AB of the triangle ABC if AC = 6 and BC = 9?

I. 3
II. 9(sqrt3)
III. 13.5
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Posts: 47924
Re: Which of the following is a possible length for side AB of triangle AB  [#permalink]

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17 Sep 2014, 00:44
2
2
jimmyjamesdonkey wrote:
Which of the following is a possible length for side AB of triangle ABC if AC = 6 and BC = 9?

I. 3
II. 9 √3
III. 13.5

(A) I only
(B) II only
(C) III only
(D) II and III
(E) I, II, and III

The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.

(9 - 6) < AB < (9 + 6)
3 < AB < 15.

Only 13.5 is in this range ($$9\sqrt{3}\approx{9*1.7}=15.3>15$$).

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Re: Which of the following is a possible length for side AB of triangle AB  [#permalink]

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23 May 2015, 23:22
Its still not convincing for me to consider C as the correct answer because no where the question states AB is the third side or the longest side.. If for example we consider the AB = 3 (answer Choice A), The side of the triangle would be 3,6 and 9.. Now 9 becomes the third side and satisfies all the constraints of the third side.. The same is applicable for the rest two choices.. I feel answer in E.. Can someone explain whats the assumption am missing here ?
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Re: Which of the following is a possible length for side AB of triangle AB  [#permalink]

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23 May 2015, 23:31
2
Hi ragunandan,

You have to be very careful with the logic here.

If you have sides of 3, 6 and 9, then you DO NOT actually have a triangle - you have a line segment with a length of 9 "on top of" another line segment with a length of 9. You can use a ruler and physically draw it on a piece of paper if that will help you to visualize the issue.

From a "math" standpoint, if 9 is the longest side, then the other two sides have to SUM to a value that is GREATER than 9 (otherwise you don't have a triangle) and a DIFFERENCE that is LESS than 9.

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Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ ***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*********************** SVP Joined: 26 Mar 2013 Posts: 1777 Re: Which of the following is a possible length for side AB of triangle AB [#permalink] ### Show Tags 23 May 2015, 23:57 EMPOWERgmatRichC wrote: Hi ragunandan, You have to be very careful with the logic here. If you have sides of 3, 6 and 9, then you DO NOT actually have a triangle - you have a line segment with a length of 9 "on top of" another line segment with a length of 9. You can use a ruler and physically draw it on a piece of paper if that will help you to visualize the issue. From a "math" standpoint, if 9 is the longest side, then the other two sides have to SUM to a value that is GREATER than 9 (otherwise you don't have a triangle) and a DIFFERENCE that is LESS than 9. GMAT assassins aren't born, they're made, Rich Hi Rich, Thanks for your nice explanations you provide but I did not understand 'you have a line segment with a length of 9 "on top of" another line segment with a length of 9. You can use a ruler and physically draw it on a piece of paper if that will help you to visualize the issue.' I tried to draw it but it is just incomplete triangle. is that the result you mean? Also I want to know if you cover those tactics and tricks for geometry in your courses. Thanks in advance for your help e-GMAT Representative Joined: 04 Jan 2015 Posts: 1894 Re: Which of the following is a possible length for side AB of triangle AB [#permalink] ### Show Tags 24 May 2015, 00:44 1 ragunandan wrote: Its still not convincing for me to consider C as the correct answer because no where the question states AB is the third side or the longest side.. If for example we consider the AB = 3 (answer Choice A), The side of the triangle would be 3,6 and 9.. Now 9 becomes the third side and satisfies all the constraints of the third side.. The same is applicable for the rest two choices.. I feel answer in E.. Can someone explain whats the assumption am missing here ? Hi ragunandan, Which of the following is a possible length for side AB of triangle ABC if AC = 6 and BC = 9? 1. It's mentioned in the question statement that AB is the side of triangle ABC. Since we have been given two sides AC and BC, AB has to the third side. 2. Secondly since the question asks the possible length of side AB, we need to consider all possible values of AB i.e. AB can be the shortest side or the longest side of triangle ABC. 3. Lastly AB = 3 is not a legible value of the 3rd side of triangle ABC with the other two sides as AC = 6 and BC = 9. In a triangle the sum of two sides should be greater than the 3rd side. In this case AB + AC = 9 = BC. Similarly AB = 15√3 is also not a legible value. Among the given options AB = 13.5 is the only value for which we can form a triangle with the other two sides being 6 and 9. Let me know in case you still have any trouble in the explanation Regards Harsh _________________ Register for free sessions Number Properties | Algebra |Quant Workshop Success Stories Guillermo's Success Story | Carrie's Success Story Ace GMAT quant Articles and Question to reach Q51 | Question of the week Must Read Articles Number Properties – Even Odd | LCM GCD Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2 Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry Algebra- Wavy line Practice Questions Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets | '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 12181 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: Which of the following is a possible length for side AB of triangle AB [#permalink] ### Show Tags 24 May 2015, 10:26 Hi Mo2men, You've discovered another variation on 'problem' of having sides of 3, 6 and 9 - if you tried to form a triangle, then the 3 and the 6 would NOT touch, so you would NOT have a triangle. This all stems from a math rule called the Triangle Inequality Theorem; it's relatively rare on the GMAT (you might not even see it on Test Day; if you do see it, you'll probably see it just once). Thus, the concept is not a big point-gainer on the GMAT, so unless you're already scoring at a really high level in the Quant, you should be focused on other subjects. To answer your last question: yes, we do cover it (and a couple of variants of it) in the EMPOWERgmat Course. GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
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Re: Which of the following is a possible length for side AB of triangle AB  [#permalink]

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04 Jun 2015, 20:50
Hi Rich, Thanks for the detailed explanation. #Helpful
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Re: Which of the following is a possible length for side AB of triangle AB  [#permalink]

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04 Jun 2015, 20:53
Hi Harsh,Thanks for the explanation.. It was simpler and neat..
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Re: Which of the following is a possible length for side AB of triangle AB  [#permalink]

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05 Jun 2015, 05:46
jimmyjamesdonkey wrote:
Which of the following is a possible length for side AB of triangle ABC if AC = 6 and BC = 9?

I. 3
II. 9 √3
III. 13.5

(A) I only
(B) II only
(C) III only
(D) II and III
(E) I, II, and III

POINT : The effort here is to make the rule of triangle more precise and less time consuming

We know the rule in different forms however we only need to know the rule in just one line

i.e. SUM OF TWO SHORTER SIDES > LONGEST SIDE OF THE TRIANGLE

FOR THE TRIANGLE TO EXIST

Let's apply this in this case

For Finding the range of values of the Third side

For minimum value of the third side AB, We must consider the given BC = 9 as the longest side

Hence AB (Min) + AC > BC

i.e. AB (Min) + 6 > 9

i.e. AB (Min) > 3

For Maximum value of the third side AB, We must consider the given BC = 9 and Ac=6 as the two shorter sides

Hence AC + BC > AB (Max.)

i.e. 6 + 9 > AB (Max.)

i.e. AB (Max.) < 15

i.e. the range of AB becomes

3 < AB < 15

I. 3 NOT ACCEPTABLE
II. 9 √3 - 9x1.73 = 15.3 NOT ACCEPTABLE
III. 13.5 ACCEPTABLE

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Re: Which of the following is a possible length for side AB of triangle AB  [#permalink]

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30 Dec 2016, 03:15
The third side of a triangle must be less than the sum of the other two sides and greater than their difference (i.e. |y - z| < x < y + z).
In this question:
|BC - AC| < AB < BC + AC
9 - 6 < AB < 9 + 6
3 < AB < 15
Only 13.5 is in this range. 9sqrt3 is approximately equal to 9(1.7) or 15.3.
The correct answer is C.
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Re: Which of the following is a possible length for side AB of triangle AB  [#permalink]

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19 Apr 2017, 22:32
jimmyjamesdonkey wrote:
Which of the following is a possible length for side AB of triangle ABC if AC = 6 and BC = 9?

I. 3
II. 9 √3
III. 13.5

(A) I only
(B) II only
(C) III only
(D) II and III
(E) I, II, and III

We can approach this problem with triangle properties in mind; though, first it is important to note that the stem does not state that this triangle is a right triangle so the Pythagorean Theorem cannot be applied:

(I) The sums of the two smaller sides of a triangle must be larger than the hypotenuse or greatest side- "3" fails this test and is therefore not a viable option
(II) If considered 9 √3 as 9(1.7) = 15.3 which is larger than 6 + 9 and thus not a viable option
(III) 13.5 is smaller than the sum of 9 and 6 and therefore a viable option

The correct answer is (C)
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Which of the following is a possible length for side AB of triangle AB  [#permalink]

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03 Jun 2017, 05:38
As pointed out earlier, the question is based on the triangle inequality theorem:
In that case AB has to be smaller than the sum of the two given sides 9+6=15 and be bigger than their difference 9-6=3
Thus you can write the inequality as follows: 3<AB<15.
If you now test the options only Roman Numeral III fits in that range, so the answer is C.
Which of the following is a possible length for side AB of triangle AB &nbs [#permalink] 03 Jun 2017, 05:38

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# Which of the following is a possible length for side AB of triangle AB

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