Every triangle

Geometry Level 5

Within a square, suppose that every third of its perimeter is connected to form a triangle. Let $x$ be the ratio of the area of the triangle to the square. If the difference of the maximum and minimum value of $x$ can be expressed in the form $\large \frac {a}{b}$ , where $a$ and $b$ are coprime positive integers, determine the value of

$3 \times (a + b)$ .

The answer is 111.

1 solution

Aditya R Mohan
Jun 10, 2015

The minimum & maximum areas will be in limiting conditions. The two limiting conditions are, when one of the vertex of the triangle is 1.) on the corner of square. 2.) mid of the square edge.

Using a graph, you can get the ratio of minimum area as 5/12 and the max as 4/9. The difference being 1/36, we get a=1;b=36.

So, 3 x (a+b) = 111.

Every triangle

The answer is 111.

1 solution

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