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Can you 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?

Squirty

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


The smaller of two similar rectangles has height $2$ units; the larger rectangle has length $6$ units.

If one rectangle has twice the area of the other, find the length of the smaller rectangle.




This problem is taken from Tony Gardiner's 'Extension Mathematics Gamma' book.
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.
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