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
