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# Area of an infinitely shaded region

## Question

Suppose you have a triangle that has 2-1 inch lengths. Divide this triangle into half by drawing a line from vertex between the two identical sides,  choose one of the sides randomly and shade it.  The non-shaded side is cut in half again.  Choose one of these sides randomly and cut it in half again shading one random piece.  If this pattern of cut, shade, cut, cut, shade, cut, cut, shade cut, cut,.... was to be continued forever, what would be the area of the shaded region?

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Spoiler

Shaded area = infinite sum {1/4 + 1/8 + 1/16 + 1/32 + ... +1/[2^(n+1)]} = 0.5

Edited by rocdocmac
numbers

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Spoiler

... if it is a right triangle. In other instances the shaded area would equal the area of the original triangle.

Please ignore my answers above - wrong, since there are two cuts in between!

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Let's try again ...

Spoiler

If a right triangle, then sum {1/4+1/16+1/64+ ...} = 1/3 inch^2,

or one-third of the area of any other triangle

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The area would be that of the original triangle,however that area would vary as determined by the length of the 3rd side

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On 11/12/2018 at 9:12 AM, Donald Cartmill said:

The area would be that of the original triangle,however that area would vary as determined by the length of the 3rd side

I agree that it will vary but we can bound the area.

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