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# Is the perimeter of a triangle greater than 1?

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Senior Manager
Joined: 22 Nov 2005
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Is the perimeter of a triangle greater than 1?  [#permalink]

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Updated on: 11 Jul 2013, 11:30
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Difficulty:

75% (hard)

Question Stats:

46% (01:38) correct 54% (01:39) wrong based on 264 sessions

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Is the perimeter of a triangle greater than 1?

(1) Two of the heights are less than 1/3
(2) One of the heights is greater than 1/2

Originally posted by gmat_crack on 10 Jun 2006, 21:59.
Last edited by Bunuel on 11 Jul 2013, 11:30, edited 2 times in total.
Edited the question and added the OA
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10 Jun 2006, 22:30
B it is
(1) is insufficient to decide whether Per > 1 or not
(2) One height >1/2 so we have at least two sides > 1/2
so the perimeter>1
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Updated on: 11 Jun 2006, 12:05
2
gmat_crack wrote:
Is the perimeter of a triangle greater than 1?

1). Two of the heights are less than 1/3
2). One of the heights is greater than 1/2

B. a side with 1/2 length makes the sum of other two sides at least > than 1/2, making P greater than 1.

Originally posted by Professor on 11 Jun 2006, 11:03.
Last edited by Professor on 11 Jun 2006, 12:05, edited 1 time in total.
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11 Jun 2006, 11:19
D. Either stmt is sufficient

Stmt1 Analysis:

If a,b,c are the three sides and h1 and h2 are the two heights (lets say h1 divides b into b' and b", and h2 divides c into c' and c".

h1 > 1/3, h2 > 1/3
(a+b') > h1
(a+b') > 1/3 ... (a)

(c+b") > h1
(c+b") > 1/3.... (b)

(a+c+b'+b") > 2/3

(a+b+c) > 2/3 which is less than 1.

Same can be done for h2.

Stmt1 is sufficient.

Stmt2 Analysis:

Using the same method as above for one height h > 1/2,

We can arrive @ (a+b+c) > 1

Stmt2 is sufficient.

Therefore D.
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11 Jun 2006, 11:21
Professor wrote:
gmat_crack wrote:
Is the perimeter of a triangle greater than 1?

1). Two of the heights are less than 1/3
2). One of the heights is greater than 1/2

B. a side with 1/2 length makes at least one of the other two sides more than 1/2, making P greater than 1.

Note that we just have to answer if (a+b+c)>1 or not.

Stmt 1 says it is not, stmt2 says it is. So separately both answer the question.

So, I think it should be D.
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11 Jun 2006, 11:26
1
Professor wrote:
gmat_crack wrote:
Is the perimeter of a triangle greater than 1?

1). Two of the heights are less than 1/3
2). One of the heights is greater than 1/2

B. a side with 1/2 length makes at least one of the other two sides more than 1/2, making P greater than 1.

I agree that answer is B but if one side is greater than 1/2 then it does not mean that one more side will be greater than 1/2. Instead, sum of other two sides will be greater than 1/2 making perimeter greater than 1/2.
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11 Jun 2006, 12:03
haas_mba07 wrote:
Professor wrote:
gmat_crack wrote:
Is the perimeter of a triangle greater than 1?

1). Two of the heights are less than 1/3
2). One of the heights is greater than 1/2

B. a side with 1/2 length makes the sum of other two sides at least > than 1/2, making P greater than 1.

Note that we just have to answer if (a+b+c)>1 or not.

Stmt 1 says it is not, stmt2 says it is. So separately both answer the question.

So, I think it should be D.

A is not suff. it doesnot tell about the third side.

if s1=0.32, s2=0.32 and s3 = 0.6, Perimeter is >1.
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11 Jun 2006, 12:13
1
ps_dahiya wrote:
Professor wrote:
gmat_crack wrote:
Is the perimeter of a triangle greater than 1?

1). Two of the heights are less than 1/3
2). One of the heights is greater than 1/2

B. a side with 1/2 length makes at least one of the other two sides more than 1/2, making P greater than 1.

I agree that answer is B but if one side is greater than 1/2 then it does not mean that one more side will be greater than 1/2. Instead, sum of other two sides will be greater than 1/2 making perimeter greater than 1/2.

let me update: I mean "a side with 1/2 length makes the sum of other two sides at least > than 1/2, making P greater than 1". right?
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Re: Is the perimeter of a triangle greater than 1? 1). Two of  [#permalink]

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11 Jul 2013, 11:07
can someone please explain the logic for this question
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Re: Is the perimeter of a triangle greater than 1? 1). Two of  [#permalink]

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11 Jul 2013, 11:54
3
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veenu08 wrote:
can someone please explain the logic for this question

STATEMENT 1:
Two heights are less than 1/3
see in figure:==> 2 triangles are there ABC and DEF
Perpendiclar from B to AC..and perpendicular from C to AB===>LESS THAN 1/3...But we can take point much higher shown....lets perpendicular from A to BC..is 100 ..SO IN THIS case perimeter you can see will be more than 1.
take second triangle DEF ===>IN this all the heights are very smalll===>you can make very small triangle whose perimeter is less than 1.
hence statement 1 is not sufficient.

STATEMENT 2:
Triangle AMC and BMC are right angled triangle with AC and BC as hypotenuses respectively. Hence, both AC and BC are greater than CM. If the height CM is greater than 1/2 then the sides AC and BC must be greater than 1/2 too. Thus the perimeter of the triangle must be greater than 1. ==>Sufficient.
hence B

HOPE IT HELPS
Attachments

tri.jpg [ 3.51 KiB | Viewed 4460 times ]

STMN 1.png [ 8.83 KiB | Viewed 4456 times ]

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Is the perimeter of a triangle greater than 1?  [#permalink]

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11 Jul 2013, 12:10
1
2
veenu08 wrote:
can someone please explain the logic for this question

Is the perimeter of a triangle greater than 1?

(1) Two of the heights are less than 1/3. Well, we can have a tiny triangle with the perimeter less than 1 but we also can have a right triangle with legs (heights) bit less than 1/3 = 0.(3). In this case hypotenuse wold be a bit less than 0.47 and the perimeter would be more than 1. Not sufficient.

(2) One of the heights is greater than 1/2. Consider the easiest case:

AB > 1/2, BC = hypotenuse > AB, thus the perimeter is for sure greater than 1. Sufficient.

Hope it's clear.

Attachment:

Perimeter2.png [ 2.22 KiB | Viewed 4514 times ]

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Re: Is the perimeter of a triangle greater than 1?  [#permalink]

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10 Dec 2013, 05:34
So in other words, you can take the height (less than 1/3 of the triangle) then take the height of the triangle formed by the new triangle (where the height is less than 1/3?) I guess there is nothing implying otherwise, but I didn't think it would count to take the height of one triangle and then another one formed inside that triangle.
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Re: Is the perimeter of a triangle greater than 1?  [#permalink]

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14 Aug 2015, 23:08
In a triangle can we conclude that length of height is always less than at least two lengths of sides ?

As we can see in following diagram length of AD is greater than length of AB and AC
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Re: Is the perimeter of a triangle greater than 1?  [#permalink]

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17 Mar 2019, 04:48
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Re: Is the perimeter of a triangle greater than 1?   [#permalink] 17 Mar 2019, 04:48
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