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If k is an integer and 2 < k < 7, for how many different : Problem Solving (PS)
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If k is an integer and 2 < k < 7, for how many different [#permalink]
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Bunuel wrote:
If k is an integer and 2 < k < 7, for how many different values of k is there a triangle with sides of lengths 2, 7, and k?

(A) one
(B) two
(C) three
(D) four
(E) five


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

So, if a triangle has sides 2, 7 and k, we can write: 7 - 2 < k < 7 + 2
Simplify to get: 5 < k < 9

We're told that k is an INTEGER, and that 2 < k < 7.
So, the only possible value of k that satisfies the inequality 5 < k < 9 is k = 6

Answer: A

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Originally posted by BrentGMATPrepNow on 12 Mar 2019, 15:53.
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Re: If k is an integer and 2 < k < 7, for how many different [#permalink]
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Bunuel wrote:
The Official Guide for GMAT® Review, 13th Edition - Quantitative Questions Project

If k is an integer and 2 < k < 7, for how many different values of k is there a triangle with sides of lengths 2, 7, and k?

(A) one
(B) two
(C) three
(D) four
(E) five

Diagnostic Test
Question: 19
Page: 22
Difficulty: 650


GMAT Club is introducing a new project: The Official Guide for GMAT® Review, 13th Edition - Quantitative Questions Project

Each week we'll be posting several questions from The Official Guide for GMAT® Review, 13th Edition and then after couple of days we'll provide Official Answer (OA) to them along with a slution.

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Believe ans is A as k can be 3,4,5 or 6. for all cases except k=6, sum of teo sides can be less than equal to third side which should not be true

As as per triangle property sum of two sides is greater than third side.
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Re: If k is an integer and 2 < k < 7, for how many different [#permalink]
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Why is it only one value?
What if 2 is the smallest side and 7 is the largest? Then, k can be 6, 7 or 8. since:

2 + 7 = 9
7 - 2 = 5
So 5 < k < 9

The problem does not specify that 2 and 7are the smaller two sides.

Originally posted by morfin on 25 Nov 2012, 14:50.
Last edited by morfin on 26 Nov 2012, 10:05, edited 1 time in total.
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Re: If k is an integer and 2 < k < 7, for how many different [#permalink]
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morfin wrote:
What if 2 is the smallest side and 7 is the largest? Then, the answer is B (two) since:

2 + 7 = 9
7 - 2 = 6
So 6 < k < 9

The problem does not specify that 2 and 7are the smaller two sides.


Not sure I understand your question.

First of all: 7-2=5, not 6.

Next, obviously since the lengths of the sides are 2, 7 and k, where 2<k<7, then the length of the smallest side is 2 and the length of the largest side is 7. Check here for complete solution: if-k-is-an-integer-and-2-k-7-for-how-many-different-135543.html#p1104032
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Re: If k is an integer and 2 < k < 7, for how many different [#permalink]
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Relationship of the Sides of a Triangle: 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.

Above relationship is in the core of this solution. Many times I have found students having difficulty is assimilating this concept.

Best way to get this concept is trying to actually draw triangles which contradict this. For example, try to draw a triangle with following sides (actual scale)

4, 3, 8

Once you failed, you will realize that 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.
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Re: If k is an integer and 2 < k < 7, for how many different [#permalink]
Bunuel wrote:
SOLUTION


Relationship of the Sides of a Triangle: 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.



Then I take it that the GMAT ignores the degenerate triangle, in which the sum of the two shorter sides can equal the longer side?
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Re: If k is an integer and 2 < k < 7, for how many different [#permalink]
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Chakolate wrote:
Bunuel wrote:
SOLUTION


Relationship of the Sides of a Triangle: 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.



Then I take it that the GMAT ignores the degenerate triangle, in which the sum of the two shorter sides can equal the longer side?


Yes, degenerate "triangle" with three collinear points is obviously not a part of GMAT quant. You should refer to OG to know what is tested on the GMAT.
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Re: If k is an integer and 2 < k < 7, for how many different [#permalink]
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Bunuel wrote:
If k is an integer and 2 < k < 7, for how many different values of k is there a triangle with sides of lengths 2, 7, and k?

(A) one
(B) two
(C) three
(D) four
(E) five


We can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle (in this case, the sides are 2 and k) must be greater than the length of its third side (in this case, 7).

Thus, we see that:

2 + k > 7

k > 5

Since k < 7, the only integer value of k that is greater than 5 but less than 7 is 6.

Answer: A
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Re: If k is an integer and 2 < k < 7, for how many different [#permalink]
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Bunuel wrote:
If k is an integer and 2 < k < 7, for how many different values of k is there a triangle with sides of lengths 2, 7, and k?

(A) one
(B) two
(C) three
(D) four
(E) five



The other two sides of the triangle can be \(5<S<9\)
So k can be only 6.

The answer is A.
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