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Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

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Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

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At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Matter of Scale

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Take any right-angled triangle with side lengths $a, b$ and $c$.

Small triangle
Make two enlargements of the triangle, by scale factors $a$ and $b$:

Enlargements
Rotate these triangles and fit them together to make a third triangle:
Enlargements fitted together
You can use the interactivity below to explore this problem. Choose a triangle, then click "New Triangle" to make copies of your triangle, which can be enlarged and rotated to fit together on the grey outline.
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Prove that the resulting triangle when the pieces are fitted together is always an enlargement of the original triangle. What is the scale factor of enlargement of the resulting triangle?

Use what you have discovered about the side lengths of the resulting triangle to come up with a proof of Pythagoras' Theorem.


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. More information on many of our other activities can be found here.
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NRICH is part of the family of activities in the Millennium Mathematics Project.