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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?

Three Cubes

Age 14 to 16 Challenge Level:
 

1. The distance between centres of two adjacent faces of a cube is $8$cm.

What is the side length of the cube?
 


 
 

2. I can make an equilateral triangle by cutting off the corner of a cube.

If the area of the largest equilateral triangle I can create in this way is $140$cm$^2$, what is the side length of this cube?






3. A third cube has an edge length of $12$cm.

At each vertex, a tetrahedron with three mutually perpendicular edges of length $4$cm is cut away.

Can you find the volume and the surface area of the remaining solid?
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