If k is an integer and 2

Question

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

Bunuel · Accepted Answer

SOLUTION

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

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.

According to the above the following must be true:  (7-2)<k<(7+2)  -->  5<k<9 . So,  k  could be 6, 7 or 8. Since also given that  2 < k < 7 , then  k=6 . Hence  k  can take only one value: 6.

Answer: A.

cyberjadugar · Answer

Hi,

Difficulty level: 600

|7-2| < k < |7+2|
or 5 < k < 9
thus k = 6, 7, 8, but 2 < k < 7
therefore, k = 6

Answer (A),

Regards,

BrentGMATPrepNow · Answer

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

RELATED VIDEO
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narangvaibhav · Answer

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.

morfin · Answer

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.

Bunuel · Answer

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 2if-k-is-an-integer-and-2-k-7-for-how-many-different-135543.html#p1104032

kundankshrivastava · Answer

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.

Chakolate · Answer

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?

Bunuel · Answer

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.

JeffTargetTestPrep · Answer

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

MHIKER · Answer

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

The answer is A.

bumpbot · Answer

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If k is an integer and 2 < k < 7, for how many different

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