Is the perimeter of triangle ABC less than 15 centimetres ?

Question

GMAT CLUB TESTS' FRESH QUESTION:

(1) The lengths of the sides of triangle ABC are consecutive even integers in centimetres
(2) The sum of the lengths of the shortest and longest sides of triangle ABC is 12 centimetres

M36-77

Bunuel · Accepted Answer

Official Solution:

Is the perimeter of triangle ABC less than 15 centimetres ? 
 
To solve this question recall the following property:  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 .
 
(1) The lengths of the sides of triangle ABC are consecutive even integers in centimetres
 
The smallest three consecutive positive even integers are 2, 4, and 6 but a triangle cannot have sides equal to 2, 4, and 6 centimetres because in this case the length of one side (6) is NOT smaller than the sum of two other sides (2 and 4). The next three consecutive positive even integers are 4, 6, and 8. The sum is 18. So, the perimeter of triangle ABC must be greater than 15 centimetres . Sufficient.
 
(2) The sum of the lengths of the shortest and longest sides of triangle ABC is 12 centimetres
 
The perimeter to be less than 15 centimetres, the third side (second smallest) must be less than 3 centimetres. In this case the sides would be: (less than 3), (less than 3), (more than 9). But since the sum of two sides is NOT greater than the third side ((less than 3) + (less than 3) < 9), than such a triangle is not possible. So, the perimeter of triangle ABC must be greater than 15 centimetres . Sufficient.

Answer: D

sumit411 · Answer

Imo d

St1: 2 is the least possible side of a triangle ( 0 is even but side with 0 units is not possible)

2,4,6-- possible pair
But as we know sum of 2 sides should be greater than 3rd side. Difference of 2 sides should be less than 3rd side.

6-2< 4 < 6+4
4< 4 < 10.. (not possible)

Next pair will always have perimeter greater than 15

Sufficient

St2: let sides be x, y, z

X+z= 12
We can infer that y should be less than 12

1+11---> 3rd side is between 10 to 12
It cannot be greater than 11 [ greatest side]. In all cases, perimeter is >15

7+5--> 3rd side is between 2 to 12
It can be but we need > or equal to 5 [ greater than smallest side]
Permimeter will always be >15

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chetan2u · Answer

Is the perimeter of triangle ABC less than 15 centimetres ?

(1) The lengths of the sides of triangle ABC are consecutive even integers in centimetres
 so the sides are 2n, 2n+2, 2n+4 
Minimum value of n is 1, so min sides are 2,4,6... But the largest side is EQUAL to sum of other two sides, so it will not be a triangle..
Next values - 4, 6, 8.....sum=4+6+8=18>15
Ans NO
Sufficient

(2) The sum of the lengths of the shortest and longest sides of triangle ABC is 12 centimetres
So if perimeter is LESS than 15, the third side can have a max value as just less than 3
The smallest side has to be less than this too, so sides are <3, <3, ~9
But such a triangle is not possible as sum of the smallest sides is LESS than the largest side.
So Ans NO
Sufficient

D

chetan2u · Answer

You are missing out on the rule that the sum of the smaller two sides has to be greater than the largest side, so statement II is also sufficient

sumit411 · Answer

Great explanation sir

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Bunuel · Answer

Par of  GMAT CLUB'S New Year's Quantitative Challenge Set

Ypsychotic · Answer

Statement 1) The lengths are consecutive even integers. Let's start by taking the shortest lengths, say, 2,4,6 perimeter=12, but these values don't form a triangle since triangle has a property that for any value of a, b and c, a+b>c. Here, 2+4= or 2+4 isn't greater than 6. Let's move forward. Take lengths as 4,6,8. Perimeter =18 and 4+6>8. For each next value, perimeter increases by 6 than the last value. for example, 6,8,10, perimeter =24. So, the perimeter is always greater than 15. Sufficient. Statement 2) Let the shortest length be a and longest be b. So, a+b =12. Let a= 5, b=7. Assume second longest be x. So, in this case, 511, x>10, y>1+11+10, y>22. So, the perimeter will be always greater than 15 cm. Hence, Sufficient. The correct answer is Option D.

exc4libur · Answer

hi  chetan2u ,

if the triangle inequality theorem refers only to the longest side, then below would not be valid? please help.

(2) The sum of the lengths of the shortest and longest sides of triangle ABC is 12 centimetres

a=shortest, c=longest

a+c=12
a=5, c=7

7-5<b<7+5
2<b<12
b=2.5

perimeter = 7+5+2.5 = 12+2.5 = 14.5 < 15…

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Is the perimeter of triangle ABC less than 15 centimetres ?

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