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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?

Interior Squares

Stage: 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Let $r$ be the radius of the circle, then for the smaller square we can apply Pythagoras' Theorem:


and for the larger square:


So $x:y = 2:5$.

This problem is taken from the UKMT Mathematical Challenges.