Similarity and congruence

There are 62 NRICH Mathematical resources connected to Similarity and congruence
Pythagoras Proofs
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Pythagoras proofs

Age
11 to 16
Challenge level
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Can you make sense of these three proofs of Pythagoras' Theorem?

Triangle in a Trapezium
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Triangle in a trapezium

Age
11 to 16
Challenge level
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Can you find and prove the relationship between the area of a trapezium and the area of a triangle constructed within it?
The square under the hypotenuse
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The square under the hypotenuse

Age
14 to 16
Challenge level
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Can you work out the side length of a square that just touches the hypotenuse of a right angled triangle?
Kite in a Square
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Kite in a square

Age
14 to 18
Challenge level
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Can you make sense of the three methods to work out what fraction of the total area is shaded?
Fit for photocopying
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Fit for photocopying

Age
14 to 16
Challenge level
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Explore the relationships between different paper sizes.
Triangle midpoints
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Triangle midpoints

Age
14 to 16
Challenge level
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You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Trapezium Four
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Trapezium four

Age
14 to 16
Challenge level
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The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
Two Ladders
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Two ladders

Age
14 to 16
Challenge level
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Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second wall. At what height do the ladders cross?
All About Ratios
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All about ratios

Age
16 to 18
Challenge level
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A new problem posed by Lyndon Baker who has devised many NRICH problems over the years.
Nicely Similar
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Nicely similar

Age
14 to 16
Challenge level
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If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?