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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
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?
At a Glance
The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?
Age 14 to 16
Challenge Level
Take any right-angled triangle with side lengths $a, b$ and $c$. For convenience, label the two acute angles $x^{\circ}$ and $y^{\circ}$.
Make two enlargements of the triangle, one by scale factor $a$ and and one by scale factor $b$:
Draw out a copy of them and indicate what the lengths and the angles are in each.
Find the lengths and angles in this last triangle.
Can you show that this triangle is similar to the original triangle?
What is the scale factor of enlargement between the first and last triangles?
Can you use your results to prove Pythagoras' Theorem?
You might like to explore some more proofs of Pythagoras' Theorem, and a proof of The Converse of Pythagoras' Theorem.
NRICH is part of the family of activities in the Millennium Mathematics Project.