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Perimeter DS

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by kop » Sun Nov 10, 2013 10:47 pm
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.

What is the answer for this?S\
I know that ST1 is sufficient, what about ST2?
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Source: — Data Sufficiency |
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by theCodeToGMAT » Sun Nov 10, 2013 11:30 pm
To find: Perimeter >30

Statement 1:
a - b = 15
that means c > 15
SUFFICIENT

Statement 2:
Area = 50
The equilateral Triangle has the minimum possible perimeter for a given Area
sqrt(3)/4 * side^2 = 50
side^2 = 50 * 4 / sqrt(3)
Side^2 = 200/1.72 = ~117
Side = 10+
So, perimeter = 30+
SUFFICIENT

Answer [spoiler]{D}[/spoiler]
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by Uva@90 » Mon Nov 11, 2013 1:00 am
Hi Kop,
This problem test the below property,

For a given Area Equilateral Triangle has the minimum possible Perimeter.

In addition there is another one too,

For a given Perimeter Equilateral Triangle has the Maximum possible Area.

Hope it helps you.

Regards,
Uva.
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by Brent@GMATPrepNow » Mon Nov 11, 2013 9:09 am
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.
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 GMATGuruNY » Mon Nov 11, 2013 11:34 am
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 > 15.
Try to MINIMIZE the values of a and b.
If b=0.01 and a=15.01, then the perimeter = a+b+c = 15.01 + 0.01+ (more than 15) = more than 30.02.
The case above illustrates that the perimeter must be greater 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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