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Prove Pythagoras' Theorem using enlargements and scale factors.

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Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC. What is the value of r/R?

Fit for Photocopying

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

This problem is available as a printable worksheet: Fit for Photocopying.pdf 

Why do this problem?

Most students will be familiar with the $A$ paper sizes but be unaware of their special properties. This problem explores ratio and scale factors, with the possibility of working with surds and indices if students choose to tackle the extension questions.

Possible approach

For students who have not yet met the relationship between length and area scale factors, Growing Rectangles offers a good introduction.
Take a sheet of $A4$ paper, hold it up to show the class.
"Here is a sheet of $A4$ paper." 

Fold it in half so that it is $A5$ size.
"Does anyone know what size this paper is?"
"What special relationships can you think of between $A4$ and $A5$ paper?"
 "The area is half"
"The long side on the $A5$ is the same as the short side on the $A4$"
"They are similar shapes"
If similarity doesn't emerge as a suggestion from the class, use the example of a photocopier - something printed on $A4$ paper can be enlarged to fit onto $A3$ without distortion, so they must be similar rectangles.
Set students the challenge of working out the scale factor of (length) enlargement needed to get from any size of paper to any other. You may wish to scaffold the task for your students by first asking them to work from $A4$ to $A2$ and then to $A3$, $A1$ and $A0$, before working out the scale factors necessary to go from larger to smaller sizes.
Finally, you may wish to spend some time with the students expressing the relationships they have discovered in a general form.
The possible extension below is suitable for students who have met negative and fractional indices. 

Key questions

When I enlarge from $A4$ to $A2$ how does the area change? How do the lengths change?
What if I went from $A(n)$ to $A(n-2)$?
When I enlarge from $A4$ to $A3$ how does the area change? How do the lengths change?
What if I went from $A(n)$ to $A(n-1)$?
What if I went from $A(n)$ to $A(n-x)$? 
What happens when I go from a larger sheet to a smaller one?

Possible extension

Starting with the definitions of the $A$ paper sizes given in the problem, challenge students to work out the dimensions first of $A0$ paper and then of $A4$ paper, which they can then check by measuring.

Possible support

Start by looking at the relationship between the even numbered paper sizes ($A4$ to $A2$, $A0$ and so on).