[phpBB Debug] PHP Notice: in file /includes/check_new_recommended_questions.php on line 37: Undefined array key "last_recommended_questions_epoch"
[phpBB Debug] PHP Notice: in file /includes/check_new_recommended_questions.php on line 41: Undefined array key "last_recommended_questions_epoch"
If the sides of a triangle have lengths x, y, and z, x + y = : Problem Solving (PS)
 Last visit was: 20 Jul 2024, 02:47 It is currently 20 Jul 2024, 02:47
Toolkit
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

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.

# If the sides of a triangle have lengths x, y, and z, x + y =

SORT BY:
Tags:
Show Tags
Hide Tags
Senior Manager
Joined: 03 Feb 2011
Status:Impossible is not a fact. It's an opinion. It's a dare. Impossible is nothing.
Affiliations: University of Chicago Booth School of Business
Posts: 469
Own Kudos [?]: 904 [67]
Given Kudos: 123
Retired Moderator
Joined: 16 Nov 2010
Posts: 903
Own Kudos [?]: 1194 [17]
Given Kudos: 43
Location: United States (IN)
Concentration: Strategy, Technology
Senior Manager
Joined: 03 Mar 2010
Posts: 257
Own Kudos [?]: 1412 [6]
Given Kudos: 22
General Discussion
Senior Manager
Joined: 03 Feb 2011
Status:Impossible is not a fact. It's an opinion. It's a dare. Impossible is nothing.
Affiliations: University of Chicago Booth School of Business
Posts: 469
Own Kudos [?]: 904 [0]
Given Kudos: 123
Brilliant !!! I was assuming there will be a solution like this. Thanks so much

There is one more thing - can you also verify the explanation in the spoiler ? Cheers

subhashghosh wrote:
(I) is out as x+y = 30 > 28 (perimeter can't be < sum of two sides)

And all answers excepy B contain I as option

Director
Joined: 08 May 2009
Status:There is always something new !!
Affiliations: PMI,QAI Global,eXampleCG
Posts: 547
Own Kudos [?]: 594 [0]
Given Kudos: 10
This took me more than 2 min's though

Using POE since x+y = 30, means 1 can be nullified altogether.

But a better approach will take a STAB at this,

I am focusing on Z (min) and Z (max) values. x+y = 30, y+z = 20 means x-z = 10

Z(min) = 1, means X = 11 and Y = 19 thus Perimeter (Z min) = 31

Z(max) = 9, since Y >10, means Y = 11 and X = 19 thus Perimeter (Zmax) = 37. Thus B fits in.

Likewise, one can try for either X(min) or Y (min) and max values too. Keeping the limits X<20,Z<30 and Y>10.
Retired Moderator
Joined: 16 Nov 2010
Posts: 903
Own Kudos [?]: 1194 [0]
Given Kudos: 43
Location: United States (IN)
Concentration: Strategy, Technology
@gmat1220, I think you're right. I also deduced x + y + z < 40 initially (by using the length of 3rd side < sum of two sides), and then I spotted the odd man out in the answer choices.
Intern
Joined: 07 Mar 2009
Status:ThinkTank
Posts: 16
Own Kudos [?]: 16 [2]
Given Kudos: 3
GPA: 3.7
1
Kudos
1
Bookmarks
The POE approach above works fast. The algebraic approach is:

First, establish the equation we are looking or x + y + z = ? and name it A

if we add both given equations we can get x + y + z + y = 50. Isolate A and you get A + y = 50

Now we know from triangle inequality theorem that x - z < y < x +z. We can get x - z by substracting both equation we are given and use the other for x + z. So we get 10 < y < 20 so:

so A = 50 - GT (10) so A = LT (40)
and A = 50 - LT (20) so A = GT (30)

30 < A < 40

Only II (36) meets this criteria.

I think you can solve under 2mn with this or even better by recognizing the trick subhashghosh explained.

Manager
Joined: 07 May 2013
Posts: 67
Own Kudos [?]: 61 [0]
Given Kudos: 1
Re: If the sides of a triangle have lengths x, y, and z, x + y = [#permalink]
20------10-------10--------40-------N
19------11-------9---------39-------Y
18------12-------8---------38-------Y
17------13-------7---------37-------Y
16------14-------6---------36-------Y
x-------y----------z------per.----triangle?
15------15-------5---------35------Y
14------16-------4---------34------Y
13------17-------3---------33------N
Math Expert
Joined: 02 Sep 2009
Posts: 94423
Own Kudos [?]: 642449 [3]
Given Kudos: 86334
Re: If the sides of a triangle have lengths x, y, and z, x + y = [#permalink]
3
Bookmarks
gmat1220 wrote:
PS : What do you think we must guess. Or is there a more intuitive approach which guarantees the result in less than 2 mins?

If the sides of a triangle have lengths x, y, and z, x + y = 30, and y + z = 20, then which of the following could be the perimeter of the triangle?

I. 28
II. 36
III. 42

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

OA is B.

I don't have the solution but this is how I think it is done. Pls verify the reasoning.

The question is basically asking us to determine the limits on x.

x + y = 30 ---- (1)
y + z = 20 ---- (2)
x - z = 10. This means y > 10 [Axiom : The third side is greater than the difference of the two sides.]

x + y = 30
y > 10
From this we get x < 20.
y + z = 20
x < 20
Adding we get x + y + z < 40 -----> I think this step is correct

From (2) we have y < 20. Since side z is non-negative. From (1) we have x > 10.
y + z = 20
x > 10
Adding we get x + y + z > 30 ------> I think this step is correct

Hence 30 < x + y + z < 40. Hence B

Similar questions to practice:
if-two-sides-of-a-triangle-have-lengths-2-and-5-which-of-th-163409.html
sam-is-training-for-the-marathon-he-drove-12-miles-from-his-158375.html
in-pqr-if-pq-x-qr-x-2-and-pr-y-which-of-the-110404.html
if-3-and-8-are-the-lengths-of-two-sides-of-a-triangular-21008.html
if-k-is-an-integer-and-2-k-7-for-how-many-different-135543.html
what-is-the-perimeter-of-isosceles-triangle-mnp-134505.html
is-the-perimeter-of-triangle-abc-greater-than-87112.html
12-easy-pieces-or-not-126366.html
in-pqr-if-pq-x-qr-x-2-and-pr-y-which-of-the-110404.html
devil-s-dozen-129312.html
you-have-6-sticks-of-lengths-10-20-30-40-50-and-133667.html
in-triangle-abc-if-ab-x-bc-y-and-ac-x-y-which-of-135495.html
what-is-the-perimeter-of-isosceles-triangle-abc-34552.html
in-triangle-abc-if-ab-x-bc-y-and-ac-x-y-which-of-135495.html

For more check Triangles chapter of Math Book: math-triangles-87197.html

Hope this helps.
Senior Manager
Joined: 13 May 2013
Posts: 311
Own Kudos [?]: 570 [1]
Given Kudos: 134
Re: If the sides of a triangle have lengths x, y, and z, x + y = [#permalink]
1
Bookmarks
PS : What do you think we must guess. Or is there a more intuitive approach which guarantees the result in less than 2 mins?

If the sides of a triangle have lengths x, y, and z, x + y = 30, and y + z = 20, then which of the following could be the perimeter of the triangle?

As others have pointed out, we can rule out I.) because it indicates that all three sides add up to 28 when the question says that just two sides add up to 30.

x + y = 30
y + z = 20

x + z + 2y = 50
We can solve by ruling out answer choices, so let's say that we assume x + y + z = 36

x + z + 2y = 50
x + y + z = 36
__________________( - )
y = 14

x + y = 30
x + (14) = 30
x = 16

x + y = 30
(16) + y = 30
y = 14

We don't even need to test III.) because it is always lumped in with I.) which we know is not possible.

B.)

I. 28
II. 36
III. 42

A I only
B II only
C I and II only
D I and III only
E I, II, and III
Manager
Joined: 25 Oct 2013
Posts: 114
Own Kudos [?]: 166 [4]
Given Kudos: 55
Re: If the sides of a triangle have lengths x, y, and z, x + y = [#permalink]
3
Kudos
1
Bookmarks
x+y=30 & y+z=20 so x+2y+z=50

x+y+z=50-y

If perimeter is 28 then y=50-28=22, and y+z=20 z cannot be negative. I is out.
If perimeter is 36 then y=50-36 = 14. z=6, x=16. no problem here.
If perimeter is 42 then y=50-42 =8. x=22, z=12. x cannot be greater than sum of y & z. III is out.

Board of Directors
Joined: 17 Jul 2014
Posts: 2145
Own Kudos [?]: 1192 [0]
Given Kudos: 236
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30
GPA: 3.92
WE:General Management (Transportation)
If the sides of a triangle have lengths x, y, and z, x + y = [#permalink]
gmat1220 wrote:
If the sides of a triangle have lengths x, y, and z, x + y = 30, and y + z = 20, then which of the following could be the perimeter of the triangle?

I. 28
II. 36
III. 42

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

my approach...
x+y=30, +z will be >30. so I is out right away. A, C, D, and E are eliminated. less than 30 seconds needed to figure it out. answer choices should be given more "confusing"...
Intern
Joined: 17 Oct 2014
Posts: 5
Own Kudos [?]: 1 [0]
Given Kudos: 109
Re: If the sides of a triangle have lengths x, y, and z, x + y = [#permalink]
Q.Integer x represents the product of all integers between 1 and 25, inclusive.
The smallest prime factor of (x + 1) must be _____.

can somebody help me how to solve this question
Math Expert
Joined: 02 Sep 2009
Posts: 94423
Own Kudos [?]: 642449 [0]
Given Kudos: 86334
Re: If the sides of a triangle have lengths x, y, and z, x + y = [#permalink]
pratistha29 wrote:
Q.Integer x represents the product of all integers between 1 and 25, inclusive.
The smallest prime factor of (x + 1) must be _____.

can somebody help me how to solve this question

This question is discussed here: integer-x-represents-the-product-of-all-integers-between-175907.html

Manager
Joined: 02 Nov 2013
Posts: 60
Own Kudos [?]: 29 [0]
Given Kudos: 10
Location: India
Re: If the sides of a triangle have lengths x, y, and z, x + y = [#permalink]
Remember the theory of the triangle,
Side of the triangle will be greater than difference of the remaining two sides and less than sum of the two remaining sides.
Let say in this case, X-Y<Z<X+Y. Looking at the answers easily we can eliminate one answer i.e. 28 which is anyway not following the first equation x + y = 30. The rest of the two options 36 is the correct answer.

My choice is B.
Manager
Joined: 20 Apr 2014
Posts: 70
Own Kudos [?]: 16 [0]
Given Kudos: 50
Re: If the sides of a triangle have lengths x, y, and z, x + y = [#permalink]
I guess the fastest and clearest approach here is to use the given choices 28 - 36 - 42.
P of triangle = x+y+z
1 ) x+y+z=28 we are given that x+y=30 so first option is false since answer can not be negative

2 ) x+y+z=36 we are given that x+y=30 so 30+z=36 z=6 y= 14 x = 16 the third side must be less than sum of other two sides and more than the difference of the other two sides of the triangle. it could be P of the triangle like that:
z<x+y z>x-y and the same for x and y

3 ) x+y+z=42 if we try the same property above, we will find that it couldn't be P of the triangle.
if it helps, please press kudos for me.
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 6804
Own Kudos [?]: 30839 [2]
Given Kudos: 799
Re: If the sides of a triangle have lengths x, y, and z, x + y = [#permalink]
2
Kudos
Top Contributor
gmat1220 wrote:
If the sides of a triangle have lengths x, y, and z, x + y = 30, and y + z = 20, then which of the following could be the perimeter of the triangle?

I. 28
II. 36
III. 42

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

First of all, the perimeter CANNOT equal 28
We know this because we're told that x + y = 30, which means the sum of two sides is 30
In order for the perimeter (x+y+z) to equal 28, side z would have to have length -2, which makes no sense.
ELIMINATE A, C, D, and E

On test day, I wouldn't spend any more time on this question.
However, let's keep going. . .

Next, we can show that the perimeter CANNOT equal 42
IMPORTANT RULE: If two sides of a triangle have lengths A and B, then . . .
DIFFERENCE between A and B < length of third side < SUM of A and B
We're told that y + z = 20, which means the sum of sides y and z is 20
The above rule tells us that the third side (side x) must be LESS THAN 20
If x is less than 20, and y+z = 20, it's impossible for the perimeter (x+y+z) to equal 42

Finally, the perimeter (x+y+z) CAN equal 36
If x = 16 y = 14, and z = 6, then all of the conditions are met, AND the perimeter is 36

RELATED VIDEO FROM OUR COURSE
Target Test Prep Representative
Joined: 14 Oct 2015
Status:Founder & CEO
Affiliations: Target Test Prep
Posts: 19175
Own Kudos [?]: 22679 [1]
Given Kudos: 286
Location: United States (CA)
Re: If the sides of a triangle have lengths x, y, and z, x + y = [#permalink]
1
Kudos
gmat1220 wrote:
If the sides of a triangle have lengths x, y, and z, x + y = 30, and y + z = 20, then which of the following could be the perimeter of the triangle?

I. 28
II. 36
III. 42

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

If we add the two equations, we have x + 2y + z = 50. Subtracting x + y + z (i.e, the perimeter of the triangle) from this, we have y = 50 - (x + y + z). Now let’s check the numbers in the given Roman numerals.

I. 28

If the perimeter is 28, then y = 50 - 28 = 22. However, it’s not possible for y + z = 20 (since z would have to be -2). Therefore, 28 can’t be the perimeter.

II. 36

If the perimeter is 36, then y = 50 - 36 = 14. In this case, x + 14 = 30 → x = 16 and 14 + z = 20 → z = 6. So we have x = 16, y = 14 and z = 6. We can see that these 3 numbers can be the side lengths of a triangle.

III. 42

If the perimeter is 42, then y = 50 - 42 = 8. In this case, x + 8 = 30 → x = 22 and 8 + z = 20 → z = 12. So we have x = 22, y = 8 and z = 12. However, these 3 numbers can’t be the side lengths of a triangle since 8 + 12 is not greater than 22.

Non-Human User
Joined: 09 Sep 2013
Posts: 34039
Own Kudos [?]: 853 [0]
Given Kudos: 0
Re: If the sides of a triangle have lengths x, y, and z, x + y = [#permalink]
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.
Re: If the sides of a triangle have lengths x, y, and z, x + y = [#permalink]
Moderator:
Math Expert
94421 posts