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# Triangle ABC has a perimeter of 18. Which of the following

tagged by: M7MBA

#### Triangle ABC has a perimeter of 18. Which of the following

Triangle ABC has a perimeter of 18. Which of the following cannot be the area of triangle ABC?

(A) 119/359
(B) 359/119
(C) π
(D) 12
(E) 16

The OA is the option E.

Experts, how can I solve this PS question? How can I know which one is the correct answer? Can you show me how would you solve it?

### GMAT/MBA Expert

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M7MBA wrote:
Triangle ABC has a perimeter of 18. Which of the following cannot be the area of triangle ABC?

(A) 119/359
(B) 359/119
(C) π
(D) 12
(E) 16
Given a FIXED PERIMETER for a triangle, the greatest possible are will be yielded if the triangle is EQUILATERAL.

The area of an equilateral triangle = (s²/4)√3.
√3 ≈ 1.7.

If a triangle with a fixed perimeter of 12 is equilateral -- implying that each side has a length of 4 -- we get:
Greatest possible area = (s²/4)√3 = (4²/4)√3 = 9√3 ≈ (9()(1.7) = 15.3.
Thus, the area cannot be equal to 16.

A similar rule:
Given a quadrilateral with a fixed perimeter, the greatest possible area will be yielded if the quadrilateral is a SQUARE.

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### GMAT/MBA Expert

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Hi M7MBA,

We're told that a triangle has a perimeter of 18. We're asked which of the following CANNOT be the area of the triangle. This question can be solved with a little logic, by TESTing VALUES and using the answer choices to our advantage.

To start, we could make LOTS of different triangles with a perimeter of 18 (keep in mind that the side lengths do NOT need to be integers) and some of the triangles would be so 'long and thin' that their areas would be almost 0. Thus, "fractional" areas are possible. Thus, we have to start thinking in terms of how big the area could get. Logically-speaking, there WILL be a maximum area, so it's likely that the answer to this question is the biggest answer. There is a way to prove it though...

Imagine if we had a 45/45/90 right triangle with sides of 5, 5 and 5√2. Since √2 = about 1.4, the perimeter of this triangle would be a little less than 18. The area of THIS triangle would be (1/2)(5)(5) = 12.5, o a slightly larger triangle would have a slightly larger area. Thus, any area less than, or equal to, 12.5 is possible. Four of the answers fit that range - and one of them does not....

GMAT assassins aren't born, they're made,
Rich

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