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Matter of Scale

Prove Pythagoras' Theorem using enlargements and scale factors.

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Conical Bottle

A right circular cone is filled with liquid to a depth of half its vertical height. The cone is inverted. How high up the vertical height of the cone will the liquid rise?

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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 builds on the ideas about length and area scale factors of enlargement introduced in Growing Rectangles.
You may be familiar with the standard paper size $A4$.

Two sheets of $A4$ fit together to make a sheet of $A3$, two sheets of $A3$ fit together to make a sheet of $A2$, and so on.

A sheet of $A0$ has an area of $1$ square metre.

Each member of the $A$ paper size family is an enlargement of the others - they are similar shapes.


What scale factor of enlargement would you need to scale the side lengths of $A4$ to $A3$?
Or $A4$ to $A2$? 
Or $A4$ to $A1$?
Or $A4$ to $A0$?

What would the scale factors be if you scaled from the larger sheets to the smaller ones?

Can you write down an expression for the linear scale factor of enlargement needed to get from $A(n)$ to $A(m)$?
You may wish to consider separately the case where $n > m$ and where $n < m$. 
On a photocopier, approximately what percentage would you need to scale by in order to photocopy an $A3$ poster onto $A4$ paper?
Here are some challenging questions to consider:
Can you express the length of the longer side of a sheet of paper from the $A$ family in terms of its shorter side?

Given that a sheet of $A0$ has an area of $1$ square metre, can you work out its dimensions?
Can you use this together with your previous results to work out the exact dimensions of a sheet of $A4$ paper?
Can you find a consistent way to define $A(-1)$ and other negative paper sizes?
Can you find a consistent way to define $A(\frac{1}{2})$, and other fractional paper sizes?