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?
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. 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?

Sitting Pretty
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Sitting pretty

Age
14 to 16
Challenge level
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A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?