M25-04

Official Solution:


Is the perimeter of a triangle with sides \(a\), \(b\), and \(c\) greater than 30 centimeters ?

(1) \(a-b=15\).

Must know for the GMAT: 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.

Hence, we have \(a+b > c\) (since the length of any side of a triangle must be less than the sum of the other two sides) and \(c > 15\) (since the length of any side of a triangle must be greater than the positive difference of the other two sides). These imply \(a+b > c > 15\). As both \(a + b\) and \(c\) exceed 15, it follows that the perimeter, \(a+b+c\), will be greater than \(15+15=30\). Sufficient.

(2) The area of the triangle is 50 centimeters .

For a given perimeter, an equilateral triangle has the largest area. Now, if the perimeter were equal to 30 centimeters, then it would have the largest area if it were equilateral.

Let's find out what this area would be: \(Area=s^2*\frac{\sqrt{3} }{4}=(\frac{30}{3})^2*\frac{\sqrt{3} }{4}=25*\sqrt{3} < 50\). Since even an equilateral triangle with a perimeter of 30 cannot have an area of 50, then the perimeter must be more than 30. Sufficient.


Answer: D