## Question

Did you forget the original question? Refresh your memory:

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

Indicate all such values

(A) 119/359

(B) 359/119

(C) π

(D) 16

## Answer and Explanation

So let’s imagine an isosceles right triangle. Remove the hypotenuse and you’ll get two line segments that meet at a right angle. Now, take the vertical segment and pull it down, the way you would a level.

What’s happening? Well, for one, the right angle is no longer a right angle, but is increasing the closer the line segment is to being horizontal. Secondly, there is going to come a point when the line is completely horizontal and the two segments know form one line. Right before this happens, when the line segment you’ve been pulling down is almost horizontal, stop and replace the hypotenuse that we unceremoniously plucked at the very beginning.

What do you notice about this triangle, besides the fact that the hypotenuse is now much longer? Well, it is one skinny triangle, an extremely skinny triangle. Not that you have to give me an exact number, but what would you say about the area? Pretty tiny, right? Well, that’s the concept. The area can be anything greater than zero, because as soon as the line becomes horizontal we no longer have a triangle—albeit a very skinny one—but a line.

Therefore, [A], [B], and [C] can all be possible answers. That’s right, π also works. See, when we “lift up” the lever, the area of the resulting triangles happens along a continuum. So anything greater than 0 (including irrational numbers, such as π) and less than ‘x’ can be valid areas. Wait a second, you’re probably thinking. ‘x’?! Well, ‘x’ stands for the great possible area of a triangle, in which the perimeter is fixed but we are free to manipulate the sides. And the kind of triangle that leads to the maximum area? An equilateral triangle. In this case, an equilateral triangle with a perimeter of 18, translates to a length of 6 per side. Using the formula for an equilateral triangle (s^2 x √3)/4) = 9√3. That gives us about 15.6, which is less than [D]. Therefore [D] __cannot__ be the area. So it is the only answer of the four that works.

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