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Perimeter of triangle

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by GMATGuruNY » Fri Sep 05, 2014 11:21 pm
kop wrote:Is the perimeter of triangle with the sides a, b and c greater than 30?
(1) a-b=15.
(2) The area of the triangle is 50.
Statement 1: a-b = 15.
The third side of a triangle must be greater than the difference between the lengths of the other 2 sides.
Thus:
c > a-b
c > 15.

Since a-b = 15, a = b+15.
Thus:
Perimeter = a + b + c = (b+15) + b + (more than 15) = 2b + (more than 30).
SUFFICIENT.

Statement 2: area = 50
Given a triangle with perimeter p, the maximum possible area will be yielded if the triangle is EQUILATERAL.
Thus, if p=30, then the maximum possible area will be yielded if each side = 10.
The area of an equilateral triangle = (s²√3)/4.
If p=30 and s=10, then the area = (10²√3)/4 = 25√3 = less than 50.
Implication:
The maximum possible area of a triangle with a perimeter of 30 is LESS THAN 50.
Since statement 2 requires that the area be EQUAL TO 50, the perimeter of the triangle must be GREATER THAN 30.
SUFFICIENT.

The correct answer is D.
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by Brent@GMATPrepNow » Sat Sep 06, 2014 6:08 am
Is the perimeter of triangle with the sides a, b and c greater than 30?

(1) a - b = 15.
(2) The area of the triangle is 50.
Target question: Is the perimeter of triangle with the sides a, b and c greater than 30?

REPHRASED target question: Is a + b + c > 30?

Statement 1: a - b = 15

IMPORTANT RULE: If two sides of a triangle have lengths A and B, then . . .
(difference of sides A and B) < third side < (sum of sides A and B)

So, (a - b) < side c < (a + b)
Replace a - b with 15 to get: 15 < side c < a + b
Since 15 < c, we can say that c = 15+ (some value greater than 15)

Also, since a - b = 15, we can say that a = b + 15

So, a + b + c = (b + 15) + b + 15+
= 2b + 30+
This means that a + b + c is definitely greater than 30
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The area of the triangle is 50
As theCodeToGMAT and Uva have stated, if we examine ALL triangles with area 50, the triangle with the shortest perimeter will be a equilateral triangle.

So, let's determine the shortest possible perimeter of a triangle with area 50.

Formula: Area of equilateral triangle = √3/4 (side)²
So, 50 = √3/4 x (side)²
Multiply both sides by 4 to get: 200 = √3(side)²
Divide both sides by √3 to get: 200/√3 = (side)²

IMPORTANT: we know that 200/2 = 100
Since √3 < 2, we know that 200/√3 > 100
In other words, 200/√3 = 100+

So, 100+ = (side)², which means side = 10+
In other words, the equilateral triangle with area 50 has sides that are each longer than 10.
In other words, the equilateral triangle with area 50 has a perimeter that's GREATER than 30

Since the perimeter is minimized when the triangle is an equilateral triangle, we can be certain that a + b + c is definitely greater than 30
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent
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by GMATinsight » Sat Sep 06, 2014 9:29 am
j_shreyans wrote:Is the perimeter of triangle with the sides a, b and c greater than 30?

(1) a−b=15

(2) The area of the triangle is 50

OAD
Perimeter of Triangle = Sum of three sides of Triangle = ???

Statement 1) a-b = 15
Property: For a triangle to Exist, SUM OF TWO SMALLER SIDES > THIRD LONGEST SIDE
Here, the least values of three sides could be b=1, a=16 and therefore c=16
Least Perimeter = 1+16+16 = 33
SUFFICIENT

Statement 2) The area of the triangle is 50
For least values of perimeter, the triangle must be equilateral
therefore, √3/4 (side)²=50
and (Side)² = 200/√3 = 200/1.73 > 100
i.e.(Side) > 10
i.e. Perimeter > 30
SUFFICIENT

Answer: Option D
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by [email protected] » Sat Apr 02, 2016 8:17 pm
Please explain it is no where given that the triangle is an equilateral triangle so why have we assumed that ..
Also it is no where stated that the perimeter needs to a least.

It is very misguiding and confusing to make such assumptions
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by GMATinsight » Sun Apr 03, 2016 4:57 am
[email protected] wrote:Please explain it is no where given that the triangle is an equilateral triangle so why have we assumed that ..
Also it is no where stated that the perimeter needs to a least.

It is very misguiding and confusing to make such assumptions
hI rOHIT,

WE ARE BASICALLY LOOKING AT THE RANGE OF POSSIBLE VALUES OF PERIMETER

If lowest perimeter too is greater that 30 for the given area then obviously higher perimeter will also be greater than 30 and the answer of the question will be consistently YES.

So for the lowest possible value of Perimeter, we need to consider the equilateral Triangle

CONCEPT TO BE NOTED HERE:
Among all triangles for given area the perimeter will be Least if the triangle is equilateral and
Among all triangles for given Perimeter the Area will be Maximum if the triangle is equilateral

There can be no assumption in GMAT and you are correct about it :P

I hope this helps!!!
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