The Fencing Problem

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The Fencing Problem

There is a need to make a fence that is 1000m long. The area inside

the fence has to have the maximum area. I am investigating which shape

would give this.

Triangles: Scalene

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The diagram above is not to scale. Instead of having the perimeter to

1000m, only in this diagram, I have made the perimeters of the shape

to 10, only to make this part of the investigation easier to

understand. We know that the base of all the shapes is 2. The lengths

for the equilateral triangle are 4 on each side. This part of the

investigation is to explain why the triangle with the longest height

cannot have the same base. The tallest triangle also has a perimeter

of 10. One of the sides for the tallest triangle is 5, which is

understandable. However the other side is 3. This is literally

impossible to be because if this triangle was drawn to scale, then the

side that is 3 will not end up reaching the base. The isosceles

triangle has a side of 4 and it looks shorter than the side of 3. The

only way the higher triangle will reach is if the base is shortened.

So in the formula 'h x b ÷ 2', in the case of the higher triangle, the

height will be longer but the base will be shorter.

Looking at this diagram, there is no need to draw out tables to find

out whether or not a scalene triangle is bigger than an equilateral or

an isosceles in terms of area. I have made it so that the base is the

same width for all triangles. The lines that are going from top to

bottom on each triangle represent the height. It shows that if the

base stays the same for all triangles, a scalene can never have the

larger height and larger base than the isosceles and if it doesn't

have the larger height and base then it cannot have the larger area

either.

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