Similarity and congruence

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Favourite

Hex

Age

11 to 14

Challenge level

Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.
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All About Ratios

Age

16 to 18

Challenge level

A new problem posed by Lyndon Baker who has devised many NRICH problems over the years.
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Figure of Eight

Age

14 to 16

Challenge level

On a nine-point pegboard a band is stretched over 4 pegs in a "figure of 8" arrangement. How many different "figure of 8" arrangements can be made ?
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Matching Triangles

Age

5 to 7

Challenge level

Can you sort these triangles into three different families and explain how you did it?
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Same Length

Age

11 to 16

Challenge level

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?
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Triangle in a Trapezium

Age

11 to 16

Challenge level

Can you find and prove the relationship between the area of a trapezium and the area of a triangle constructed within it?
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Pythagoras Proofs

Age

11 to 16

Challenge level

Can you make sense of these three proofs of Pythagoras' Theorem?
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Making Sixty

Age

14 to 16

Challenge level

Why does this fold create an angle of sixty degrees?
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Favourite

Triangle Midpoints

Age

14 to 16

Challenge level

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?
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Two Ladders

Age

14 to 16

Challenge level

Two ladders are propped up against facing walls. At what height do the ladders cross?