GMAT PREP PS question

alex.gellatly wrote:If two sides of a triangle have lengths 2 and 5, which of the following could be the perimeter of the triangle?

I 9
II 15
III 19

(A) None
(B) I only
(C) II only
(D) II and III only
(E) I, II, and III

Hi!

This question is testing you on the "length of a side of a triangle rule":

each side must be greater than the positive difference between the other two sides and less than the sum of the other two sides, or, if we call the sides x, y and z:

|y-z| < x < y + z

Applying that rule to this question, we can quickly see that our 3rd side must be less than 7. Since 2+5+7 is 14, the perimeter must be less than 14.

Now we have to check the bottom limit: x has to be greater than the positive difference of the other two sides, so x must be greater than 5-2 = 3. 5+2+3 = 10, so x must be greater than 10.

Since there are no answers between 10 and 14, choose A!

For all Triangles,
Difference between two sides < other side < Sum of two sides
5 - 2 < a < 5 + 2 (where 'a' is the length of the third side)
3 < a < 7
adding (5 + 2) on all the three sides of the above inequality,
10 < Perimeter < 14

None of the values given in I, II and III are possible.

(A) is the answer.

By using the property of triangle that sum of two sides should be greater than the third side, we can say that

third side should be less than 7 (since 5+2 should be greater than 3rd side) and it should be greater than 3( since 3rd side+2 should be greater than 5)

So 3rd side can be 4, 5 Or 6

So possible perimeters can be
i) 5+4+2 = 11
ii)5+5+2 = 12
iii) 5+6+2 = 13

None is given in the options provided, Hence choice A

Can it also be said that the Sum of 2 smaller sides must be greater then the bigger side?
In this case, taking the possible values for the other side into consideration:
2 + 2 < 5
2 + 5 < 8
2 + 5 < 12
Hence answer is A.

Anurag@Gurome wrote:

alex.gellatly wrote:If two sides of a triangle have lengths 2 and 5, which of the following could be the perimeter of the triangle?

I 9
II 15
III 19

(A) None
(B) I only
(C) II only
(D) II and III only
(E) I, II, and III

Sum of two sides of a triangle must be greater than the third side.
So, with lengths of sides 2 and 5, there can be 3 different length for the third side: 6, 5, 4 (not 3 because 2 + 3 = 5 not greater than 5)
So, perimeters of 11, 12, 13 are possible.
None of the above.

The correct answer is A.