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Prove Pythagoras' Theorem using enlargements and scale factors.

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?


Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

Similar Rectangles

Age 14 to 16
Challenge Level

Call the length of the smaller rectangle $x$.

If the area of the smaller rectangle is $2x$, the area of the larger rectangle is $4x$, and the height of the larger rectangle is $4x/6$. The height:length ratio of both rectangles must be the same, since they are similar.

So $\frac{2}{x} = \frac{4x}{36}$
$x^{2} = 18$

Alternatively, call the height of the larger rectangle $y$.

Then, comparing the areas, $4x = 6y$

By similarity, $\frac{y}{2} = \frac{6}{x}$
i.e. $y = \frac{12}{x}$

Then by substitution, $4x = \frac{72}{x}$
$x^{2} = 18$
$x = \sqrt{18}$