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

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?

Right Angled Possibilities

Stage: 4 Short Challenge Level:

The side of length 5cm cannot be the hypotenuse of the right-angled triangle as it is shorter than the side of length 6cm. If the 6cm side is the hypotenuse, then the third side of the triangle has length $\sqrt{11}$cm. If the 6cm side is not the hypotenuse, then the hypotenuse has length $\sqrt{61}$cm. These are the only two possibilities.

This problem is taken from the UKMT Mathematical Challenges.
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