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Adding Triangles

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Take an A4 piece of paper and halve it by drawing a line across the middle, parallel to the shorter side. Each half is called an A5 rectangle.
Halve the bottom half by drawing a vertical line down the middle. This creates two A6 rectangles.
Halve the right hand one by drawing a horizontal line across its middle. This creates two A7 rectangles.
Halve the bottom one by drawing a vertical line down the middle. This creates two A8 rectangles.
You should have something like this:

paper divided as instructed

Halve the right hand A8 shape by drawing a horizontal line across its middle. This creates two A9 rectangles.
And so on ... Keep going until the rectangles are too small to be seen clearly.

Now draw the diagonal of the A4 piece of paper from the top left corner to the bottom right corner. This creates a sequence of triangles. The first two are numbered in the diagram below but you will have many more drawn on your sheet:

as above with diagonal drawn in showing first two triangles

What is the total area of the first two triangles as a fraction of the original A4 rectangle?
What is the total area of the first three triangles as a fraction of the original A4 rectangle?
If you could you go on adding all the triangles' areas, what do you think the total would be as a fraction of the original A4 rectangle?

Many thanks to Professor Michael Sewell for providing us with this idea.
It has a connection with Zeno's paradox. Can you find out about that paradox?
What is the connection between the method used to find all the triangles' areas and the method used to explain Zeno's paradox?